A lawyer for former adult-film star Stephanie Clifford has frustrated efforts by federal prosecutors to obtain information about a hush-money deal involving President Donald Trump’s personal lawyer, according to people familiar with the matter.

In public, Michael Avenatti, Ms. Clifford’s current attorney, has been among the most vocal critics of Michael Cohen, the lawyer who paid her $130,000 in October 2016 to sign a nondisclosure agreement about an alleged 2006 sexual encounter with Mr. Trump.

But...