diff --git a/src/components/views/context_menus/MessageContextMenu.tsx b/src/components/views/context_menus/MessageContextMenu.tsx
index cf61ee5bfd..86e9d9cc30 100644
--- a/src/components/views/context_menus/MessageContextMenu.tsx
+++ b/src/components/views/context_menus/MessageContextMenu.tsx
@@ -70,8 +70,8 @@ interface IProps extends IPosition {
     rightClick?: boolean;
     // The Relations model from the JS SDK for reactions to `mxEvent`
     reactions?: Relations;
-    // A permalink to the event
-    showPermalink?: boolean;
+    // A permalink to this event or an href of an anchor element the user has clicked
+    link?: string;
 
     getRelationsForEvent?: GetRelationsForEvent;
 }
@@ -227,7 +227,7 @@ export default class MessageContextMenu extends React.Component<IProps, IState>
         this.closeMenu();
     };
 
-    private onPermalinkClick = (e: React.MouseEvent): void => {
+    private onShareClick = (e: React.MouseEvent): void => {
         e.preventDefault();
         Modal.createTrackedDialog('share room message dialog', '', ShareDialog, {
             target: this.props.mxEvent,
@@ -236,9 +236,9 @@ export default class MessageContextMenu extends React.Component<IProps, IState>
         this.closeMenu();
     };
 
-    private onCopyPermalinkClick = (e: ButtonEvent): void => {
+    private onCopyLinkClick = (e: ButtonEvent): void => {
         e.preventDefault(); // So that we don't open the permalink
-        copyPlaintext(this.getPermalink());
+        copyPlaintext(this.props.link);
         this.closeMenu();
     };
 
@@ -295,11 +295,6 @@ export default class MessageContextMenu extends React.Component<IProps, IState>
         });
     }
 
-    private getPermalink(): string {
-        if (!this.props.permalinkCreator) return;
-        return this.props.permalinkCreator.forEvent(this.props.mxEvent.getId());
-    }
-
     private getUnsentReactions(): MatrixEvent[] {
         return this.getReactions(e => e.status === EventStatus.NOT_SENT);
     }
@@ -318,11 +313,11 @@ export default class MessageContextMenu extends React.Component<IProps, IState>
     public render(): JSX.Element {
         const cli = MatrixClientPeg.get();
         const me = cli.getUserId();
-        const { mxEvent, rightClick, showPermalink, eventTileOps, reactions, collapseReplyChain } = this.props;
+        const { mxEvent, rightClick, link, eventTileOps, reactions, collapseReplyChain } = this.props;
         const eventStatus = mxEvent.status;
         const unsentReactionsCount = this.getUnsentReactions().length;
         const contentActionable = isContentActionable(mxEvent);
-        const permalink = this.getPermalink();
+        const permalink = this.props.permalinkCreator?.forEvent(this.props.mxEvent.getId());
         // status is SENT before remote-echo, null after
         const isSent = !eventStatus || eventStatus === EventStatus.SENT;
         const { timelineRenderingType, canReact, canSendMessages } = this.context;
@@ -420,17 +415,13 @@ export default class MessageContextMenu extends React.Component<IProps, IState>
         if (permalink) {
             permalinkButton = (
                 <IconizedContextMenuOption
-                    iconClassName={showPermalink
-                        ? "mx_MessageContextMenu_iconCopy"
-                        : "mx_MessageContextMenu_iconPermalink"
-                    }
-                    onClick={showPermalink ? this.onCopyPermalinkClick : this.onPermalinkClick}
-                    label={showPermalink ? _t('Copy link') : _t('Share')}
+                    iconClassName="mx_MessageContextMenu_iconPermalink"
+                    onClick={this.onShareClick}
+                    label={_t('Share')}
                     element="a"
                     {
                         // XXX: Typescript signature for AccessibleButton doesn't work properly for non-inputs like `a`
                         ...{
-
                             href: permalink,
                             target: "_blank",
                             rel: "noreferrer noopener",
@@ -508,6 +499,26 @@ export default class MessageContextMenu extends React.Component<IProps, IState>
             );
         }
 
+        let copyLinkButton: JSX.Element;
+        if (link) {
+            copyLinkButton = (
+                <IconizedContextMenuOption
+                    iconClassName="mx_MessageContextMenu_iconCopy"
+                    onClick={this.onCopyLinkClick}
+                    label={_t('Copy link')}
+                    element="a"
+                    {
+                    // XXX: Typescript signature for AccessibleButton doesn't work properly for non-inputs like `a`
+                        ...{
+                            href: link,
+                            target: "_blank",
+                            rel: "noreferrer noopener",
+                        }
+                    }
+                />
+            );
+        }
+
         let copyButton: JSX.Element;
         if (rightClick && getSelectedText()) {
             copyButton = (
@@ -566,10 +577,11 @@ export default class MessageContextMenu extends React.Component<IProps, IState>
         }
 
         let nativeItemsList: JSX.Element;
-        if (copyButton) {
+        if (copyButton || copyLinkButton) {
             nativeItemsList = (
                 <IconizedContextMenuOptionList>
                     { copyButton }
+                    { copyLinkButton }
                 </IconizedContextMenuOptionList>
             );
         }
diff --git a/src/components/views/rooms/EventTile.tsx b/src/components/views/rooms/EventTile.tsx
index e8f0199c5f..e54c0900a5 100644
--- a/src/components/views/rooms/EventTile.tsx
+++ b/src/components/views/rooms/EventTile.tsx
@@ -234,7 +234,7 @@ interface IState {
     // Position of the context menu
     contextMenu?: {
         position: Pick<DOMRect, "top" | "left" | "bottom">;
-        showPermalink?: boolean;
+        link?: string;
     };
 
     isQuoteExpanded?: boolean;
@@ -842,26 +842,27 @@ export class UnwrappedEventTile extends React.Component<IProps, IState> {
     };
 
     private onTimestampContextMenu = (ev: React.MouseEvent): void => {
-        this.showContextMenu(ev, true);
+        this.showContextMenu(ev, this.props.permalinkCreator?.forEvent(this.props.mxEvent.getId()));
     };
 
-    private showContextMenu(ev: React.MouseEvent, showPermalink?: boolean): void {
+    private showContextMenu(ev: React.MouseEvent, permalink?: string): void {
+        const clickTarget = ev.target as HTMLElement;
+
         // Return if message right-click context menu isn't enabled
         if (!SettingsStore.getValue("feature_message_right_click_context_menu")) return;
 
-        // Return if we're in a browser and click either an a tag or we have
-        // selected text, as in those cases we want to use the native browser
-        // menu
-        const clickTarget = ev.target as HTMLElement;
-        if (
-            !PlatformPeg.get().allowOverridingNativeContextMenus() &&
-            (clickTarget.tagName === "a" || clickTarget.closest("a") || getSelectedText())
-        ) return;
+        // Try to find an anchor element
+        const anchorElement = (clickTarget instanceof HTMLAnchorElement) ? clickTarget : clickTarget.closest("a");
 
         // There is no way to copy non-PNG images into clipboard, so we can't
         // have our own handling for copying images, so we leave it to the
         // Electron layer (webcontents-handler.ts)
-        if (ev.target instanceof HTMLImageElement) return;
+        if (clickTarget instanceof HTMLImageElement) return;
+
+        // Return if we're in a browser and click either an a tag or we have
+        // selected text, as in those cases we want to use the native browser
+        // menu
+        if (!PlatformPeg.get().allowOverridingNativeContextMenus() && (getSelectedText() || anchorElement)) return;
 
         // We don't want to show the menu when editing a message
         if (this.props.editState) return;
@@ -875,7 +876,7 @@ export class UnwrappedEventTile extends React.Component<IProps, IState> {
                     top: ev.clientY,
                     bottom: ev.clientY,
                 },
-                showPermalink: showPermalink,
+                link: anchorElement?.href || permalink,
             },
             actionBarFocused: true,
         });
@@ -924,7 +925,7 @@ export class UnwrappedEventTile extends React.Component<IProps, IState> {
                 onFinished={this.onCloseMenu}
                 rightClick={true}
                 reactions={this.state.reactions}
-                showPermalink={this.state.contextMenu.showPermalink}
+                link={this.state.contextMenu.link}
             />
         );
     }
diff --git a/src/i18n/strings/en_EN.json b/src/i18n/strings/en_EN.json
index 4c61d944e8..aa4dcd2cdd 100644
--- a/src/i18n/strings/en_EN.json
+++ b/src/i18n/strings/en_EN.json
@@ -2921,10 +2921,10 @@
     "Forward": "Forward",
     "View source": "View source",
     "Show preview": "Show preview",
-    "Copy link": "Copy link",
     "Source URL": "Source URL",
     "Collapse reply thread": "Collapse reply thread",
     "Report": "Report",
+    "Copy link": "Copy link",
     "Forget": "Forget",
     "Mentions only": "Mentions only",
     "See room timeline (devtools)": "See room timeline (devtools)",