Cohen was invited to appear before the House oversight committee on Feb. 7.
Cohen attorney Lanny Davis said in a statement verbal attacks against Cohen and his family -- by Trump and Rudy Giuliani, Trump's current personal lawyer -- were the reasons for the delay. Davis said some were made as recently as last weekend.
Davis didn't give a specific date for Cohen to appear, saying only it would be a "later date."
Responding to the announcement, Trump accused Cohen of being "threatened by the truth."
"He has other clients also, I assume, and he doesn't want to tell the truth for me or" his other clients, he said.
After an investigation by special counsel Robert Mueller, Cohen, a former executive in the Trump Organization, pleaded guilty in August to eight counts of campaign finance violations, bank fraud and tax fraud. He received a fine and a three-year prison sentence.