(CNN) President Donald Trump's former personal lawyer Michael Cohen was subpoenaed Thursday to testify before the Senate Intelligence Committee in mid-February, a source close to Cohen told CNN.

Cohen's lawyer Lanny Davis confirmed the subpoena was issued Thursday and said that Cohen intended to comply. "We will comply and hope to agree upon reasonable terms, ground rules and a date," Davis said.

Still, the source said Cohen has the same concerns regarding the safety of his family that led him to postpone his scheduled public appearance before the House Oversight Committee next month.

Senate Intelligence traditionally does their interviews behind closed doors, not publicly.

Sen. Angus King, a Maine independent who's a member of the Intelligence Committee, told CNN's Wolf Blitzer on "The Situation Room" that a Cohen interview by the committee would be behind closed doors.

Senate Intelligence Chairman Richard Burr, a North Carolina Republican, declined to comment on the subpoena. A spokeswoman for the committee also declined to comment.

The congressional subpoena is the first of potentially several that Cohen could face before he reports to prison on March 6 for his three-year jail sentence after pleading guilty to tax and campaign finance crimes, as well as lying to Congress in his 2017 congressional testimony.

House Oversight Chairman Elijah Cummings of Maryland and House Intelligence Chairman Adam Schiff of California both said this week that they expect Cohen to testify before their panels — House Oversight in public and House Intelligence behind closed doors — and could also issue subpoenas to compel Cohen's appearance.

Virginia Sen. Mark Warner, the top Democrat on the committee, declined to comment on the subpoena, but said it was essential that Cohen testify again.

"Mr. Cohen was someone who lied to the committee, that got him into real jeopardy. ... He has an enormous number of unanswered questions about Trump Tower and a variety of other items that we need answers to," Warner said.

Cohen was scheduled to appear before the Oversight Committee on February 7 voluntarily, but he announced Wednesday he was postponing that appearance, citing threats to his family from Trump and Trump's attorney Rudy Giuliani.

"Due to ongoing threats against his family from President Trump and Mr. Giuliani, as recently as this weekend, as well as Mr. Cohen's continued cooperation with ongoing investigations, by advice of counsel, Mr. Cohen's appearance will be postponed to a later date," Davis said in a statement Wednesday.

Trump suggested in interviews and on Twitter that Cohen cut a deal to protect his wife and father-in-law, though he hasn't offered any evidence of those claims. On Wednesday, he dismissed Cohen's concerns about threats. "Well, I would say he's been threatened by the truth," Trump told reporters.

Cummings told reporters on Wednesday that he believed Cohen was acting in good faith with the committee, and had a general concern for his family's safety. But he vowed that Cohen would ultimately appear, though he did not say whether he would issue a subpoena.

"I promise you that we will hear from Mr. Cohen," Cummings said. "We will make those determinations soon and we will let you know how we plan to proceed, but we will get the testimony as sure as night becomes day and day becomes night."

Cummings said that he felt he could bring in Cohen even after he reports to prison, but Schiff said he thought it was necessary for Cohen to appear before March 6.

"We've been in discussions with him and he understands that we'll be wanting him before our committee. We hope to get his testimony voluntarily, but if necessary we'll use a subpoena," Schiff said Wednesday.

Cohen's testimony before the Oversight and Intelligence panels would cover different topics. An Oversight Committee hearing would be public, which means that Cohen would not testify on any matters concerning the ongoing special counsel's investigation into Russia — and topics like the Trump Tower Moscow project and the payments made during the campaign to women could be off-limits, too.

If Cohen doesn't want to testify, it's not clear what legal recourse he would have to avoid appearing after receiving a subpoena.

Former federal prosecutors say Cohen would have to testify about the crimes he's pleaded guilty to, including lying to Congress and campaign finance violations, because he waived any right against self-incrimination as part of those plea agreements. But he may be able to argue that he can't answer questions about other topics "if he has a fear of (prosecution for) other things for which he has not been convicted," one former prosecutor said.

CNN legal analyst Michael Zeldin explained: "As to matters relating to his plea agreements Cohen likely retains no Fifth Amendment right. With respect to collateral matters (for example: 'What do you know about this murder?') he arguably retains his Fifth Amendment rights, but it's hard to imagine how this would arise in connection with his congressional testimony."

Zeldin said that means Cohen would likely have to answer questions about the Trump Tower Moscow deal, because that's what he pleaded guilty to lying to Congress about. "I would expect that he waived his rights with respect to this topic," he said. "But, if there are areas that go beyond his previous testimony and as to which he may still be under investigation by Mueller, he may have a colorable claim of privilege. It's complicated."

Because Cohen has already appeared before several congressional committees, Zeldin added, "his previous testimony before Congress itself may have constituted a waiver of his Fifth Amendment rights." Cohen appeared behind closed doors separately before the House and Senate Intelligence committees in 2017. In December, he pleaded guilty to lying to Congress about how long the Trump Tower Moscow project discussions extended into the 2016 campaign.

Schiff says he wants Cohen to testify behind closed doors so he can be asked questions about the Russia investigation.

It's likely Burr also wants Cohen to appear behind closed doors as part of the panel's bipartisan investigation into Russia's 2016 election meddling that's lasted for roughly two years.

In the wake of his guilty plea, Burr said he was confident that he would bring in Cohen before he reported for prison.

This story has been updated with additional developments from Thursday.