By Jeremy Herb and Kara Scannell, CNN

(CNN) -- President Donald Trump's former personal lawyer Michael Cohen has agreed to testify behind closed doors before the House Intelligence Committee next week -- and he's tentatively agreed to go through with his planned testimony before the House Oversight Committee, according to Cohen's new lawyer Michael Monico.

Cohen's Oversight Committee testimony is not yet finalized, but he's in discussions to appear on February 7, the date he was initially scheduled to publicly testify before he abruptly announced last week that he was postponing his appearance.

House Intelligence Chairman Adam Schiff, a California Democrat, announced Monday that Cohen had agreed to testify before the intelligence panel behind closed doors on February 8.

Cohen has also been subpoenaed by the Senate Intelligence Committee to appear behind closed doors on February 12, according to his attorney.

Monico said Cohen wants to cooperate with all of the congressional inquiries. But the former Trump attorney's appearances on Capitol Hill appeared to be off last week when he announced he was postponing due to "ongoing threats against his family" from Trump and the President's attorney, Rudy Giuliani.

Lawmakers, however, made clear they did not plan to let Cohen avoid any congressional testimony before he reports for a three-year prison sentence on March 6 after pleading guilty to campaign finance violations, tax crimes and lying to Congress.

Senate Intelligence leaders subpoenaed Cohen a day after he announced the postponement, and both Oversight Chairman Elijah Cummings of Maryland and Schiff said they would do whatever they needed to so Cohen would testify.

"We will continue to work with Mr. Cohen and law enforcement in order to protect Mr. Cohen and his family," Schiff said in a statement Monday, in which he thanked Cohen for agreeing to talk to the intelligence panel voluntarily.

Trump and Giuliani have dismissed the accusation that they were threatening Cohen.

If Cohen ultimately appears before all three congressional panels, he is likely to be speaking about different subjects publicly and privately. Before he postponed, Cummings had said that Cohen would not be speaking in an open hearing about the ongoing investigation by special counsel Robert Mueller.

Schiff has argued that Cohen needed to appear before the House Intelligence Committee, too, so he could be quizzed about Russia behind closed doors.

Cohen already testified in 2017 before both the House and Senate Intelligence committees. That testimony led to his December guilty plea for lying to Congress about how long the discussions surrounding the Trump Tower Moscow project extended into the 2016 presidential campaign.

This story has been updated with additional developments Monday.