BRUSSELS—In a bid to win European Union antitrust approval of its $26 billion acquisition of LinkedIn Corp., Microsoft Corp. has proposed concessions that would allow rival professional social networks access to its Outlook programs, according to people familiar with the matter.

After the merger, the people said, rival networks would have access to Microsoft’s Outlook add-ins program—which allows third-party tools to integrate with Outlook—and Outlook APIs—or application programming interface, which allows discrete programs...