Cohen has connected Trump to campaign-finance violations. He said Trump directed him to handle hush-money payments shortly before the 2016 election to porn star Stormy Daniels and Playboy model Karen McDougal to keep them from discussing alleged affairs with Trump from years earlier.
Trump has denied the affairs.
Lanny Davis, Cohen's lawyer and spokesman, said in a statement that, "Rather than continuing to attack Michael Cohen and his family, ignoring warnings from congressional leadership, on International Women's Day the president of the United States should instead be explaining why he, his son and CFO signed checks for hush money to keep a woman quiet."
Neither the White House nor Sperry immediately responded to CNBC's requests for comment.
Cohen is due to begin a three-year prison term on May 6.
Trump and his current lawyer Rudy Giuliani have previously mentioned Cohen's father-in-law, Fima Shusterman, when they have criticized Cohen.
Trump in January had urged people to "watch" Cohen's father-in-law, without identifying him by name.
Giuliani told CNN the same month without evidence that Shusterman may be tied to organized crime. The former New York City mayor and federal prosecutor also claimed that Shusterman was involved in criminal activity with Cohen, and that Cohen is withholding information about that activity because testifying about it would be dangerous for his father-in-law.