A closed-door interview before the House Intelligence committee was then postponed "in the interests of the investigation", Chairman Adam Schiff said.
Prosecutors did not object to a "one-time" 60-day delay from the original date Cohen was supposed to report to prison, March 6, Cohen's lawyers said.
Cummings took over the Oversight Committee after Democrats gained control of the House in November's mid-term elections.
To those who are "whipped up" about the Times' story, McCarthy noted that Special Counsel Robert Mueller hasn't "returned a single sentence that suggests that Trump committed any misconduct".
Michael Cohen, President Trump's former personal attorney, will testify before the House Oversight Committee next week, the panel announced Wednesday night.
The probe focused on hush money Cohen paid out to women during the election campaign who claimed to have had affairs with Trump.
"Mr Cohen will only talk about personal experiences and anecdotes over his 10 years working for Mr Trump", Davis said, adding that his client would speak to why he can "no longer defend" Trump and sees him as a threat to the country.
"Mr. Cohen underwent serious shoulder surgery and this extra time allows Mr. Cohen to continue his physical therapy", Cohen's legal team said in an emailed statement.
The U.S. Attorney's Office for the Northern District of California filed charges against the analyst, John Fry, for the disclosure of a suspicious activity report, which banks file on possible illegal transactions. He told lawmakers he had abandoned the project in January 2016, but he actually continued pursuing it for months after that. His congressional testimony has been rescheduled for next Wednesday, Feb. 27.
Monico also said the attorneys for the Southern District of NY, where Cohen's case has played out, did not object to the extension. "In addition, he will be able to prepare for the expected testimony next week before congressional committees, which he welcomes".