With reporting by Paula Reid

President Trump's former personal lawyer, Michael Cohen, has received a congressional subpoena to testify before the Senate Intelligence Committee, CBS News has confirmed. A source familiar with the subpoena confirmed it is for a closed hearing with the committee. Cohen's lawyer, Lanny Davis, told CBS News' Paula Reid in an interview Thursday that while the ground rules still need to be discussed, "Of course, Mr. Cohen intends to comply" with the subpoena.

The subpoena comes the day after Cohen postponed his planned testimony before the House Oversight Committee on Feb. 7, citing "ongoing threats" by President Trump and his lawyer Rudy Giuliani.

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Davis is now calling for an immediate House censure of Mr. Trump, claiming he conducted obstruction and tampering with a congressional witness -- and a federal criminal investigation and possible indictment of Giuliani. Davis accused the president of witness tampering, not only because of the statements he's made that have been critical of Cohen, "but by attacking his family, which has nothing to do with whatever he has against Mr. Cohen for telling the truth," he told Reid.

Mr. Trump referred to Cohen's family in a tweet on Friday, saying that Cohen was "lying to reduce his jail time" and ended the tweet with "watch father-in-law!" Cohen was sentenced in December to three years in federal prison, followed by three years of supervised release, stemming from his guilty plea to campaign finance violations related to payments to women who claimed to have had affairs with Mr. Trump.

A few days ago, Giuliani suggested Cohen's father-in-law might have mob connections, saying on CNN's "State of the Union," "He may have ties to something called organized crime." He also said of Cohen's father-in-law, "The man was involved in criminal activity with Michael Cohen."

Earlier this month, Cohen agreed to testify before the House Oversight Committee. Committee Chairman Elijah Cummings said in a statement at the time that the Democratic majority in the House showed that the public wants Congress "to serve as an independent check and balance on the Executive Branch by restoring accountability and transparency," and that Cohen's testimony would be a part of that mission.

Cummings told reporters Wednesday that Cohen's sense of being threatened by the president is "simply unacceptable in a democracy," and he vowed that the committee would hear from Cohen.

The announcement comes after the publication of a controversial BuzzFeed report that said Mr. Trump had instructed his longtime personal lawyer Michael Cohen to lie to Congress about the Moscow Trump Tower talks, a report refuted by the special counsel's office.

Richard Escobedo contributed to this report.