Michael Cohen initially balked at the idea of buying the silence of a former adult-film star who says she had sex with Donald Trump, but he did an about-face after a video of Mr. Trump talking about groping women became public in October 2016.

A day after the recording surfaced of outtakes of Mr. Trump speaking to a host of NBC’s “Access Hollywood,” Mr. Cohen, then Mr. Trump’s senior counsel, told a representative for the performer that he was open to a deal, according to a person familiar with the conversation.

Within...