Michael D. Cohen, President Trump’s former personal lawyer who implicated him in a scheme to pay hush money to two women claiming to have had affairs with him, has agreed to testify before the House Oversight Committee next month and give “a full and credible account” of his work for Mr. Trump.

Mr. Cohen’s decision to appear before the House Oversight and Reform Committee on Feb. 7 sets the stage for a blockbuster public hearing that threatens to further damage the president’s image and could clarify the depth of his legal woes. Mr. Cohen, a consigliere to Mr. Trump when he was a real estate developer and presidential candidate as well as informally as president, was privy to the machinations of Mr. Trump’s inner circle and key moments under scrutiny by both the special counsel, Robert S. Mueller III, and federal prosecutors in New York.

He could soon share them on national television under oath.