Cohen acknowledged in the Mueller investigation that he lied to Congress by saying negotiations over a Trump Tower in Moscow had ended in January 2016 when he actually pursued the project into that June, well into Trump's 2016 presidential campaign.
Cohen is set to testify next month before the House Oversight and Reform Committee chaired by Rep. Elijah Cummings, a Maryland Democrat.
Cohen pleaded guilty in two separate cases for campaign finance violations, tax evasion and lying to Congress.
Cohen's cooperation with Mueller prompted a falling out with his former client, with Trump attacking Cohen on Twitter, including accusing him of lying to try to lessen his jail sentence.
When asked by a reporter about Cohen's upcoming testimony during his visit to the southern border on Thursday, Trump said he isn't concerned. Cohen has said he made the payments at Trump's direction and with the intent to influence the election. Trump has denied the trysts.
Committee Chairman Elijah Cummings (D-MD) was thankful Cohen came forward on his own accord.
Mr Trump acted because he "was very concerned about how this would affect the election", Cohen said.
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Derbyshire Police confirmed on Friday that "a man was found at the perimeter fence of Moor Farm and was spoken to". But when it becomes public it doesn't provoke the same indignation as it does in England.
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The estimate is slightly higher than the 266.3 billion barrel figure previously published by the Saudi government. Saudi Arabia is the world's top oil exporter.
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I didn't say these words to people, at the time , this was our dumb asses on Twitter going back and forth with each other. He said Wednesday that he does not have an issue with the Academy. "I think it needs to be somebody who loves the films".
Trump lashed out at Cohen after his sentencing last month, tweeting that his former close confidant only agreed to plead guilty "in order to embarrass the president and get a much reduced prison sentence, which he did".
On Thursday, Cummings said he was consulting Mueller's office to ensure that he did not hinder its efforts.
Cummings has signalled that his committee is more interested in investigating the president's involvement in the campaign violations to which Cohen pleaded guilty past year.
The Oversight Committee apparently isn't the Capitol Hill body with which Cohen is eager to speak.
He added, "As a lawyer, Mr. Cohen should have known better".
The public release of a redacted transcript of his previous closed-door testimony to the House Intelligence Committee in its own Russian Federation investigation is expected to be released soon, along with the testimony of dozens of other witnesses in that probe.
Rep. Adam Schiff, D-Calif., who is the new chairman of the House Intelligence Committee, told NBC News it would also be essential that "Cohen to answer questions pertaining to the Russian Federation investigation, and we hope to schedule a closed session before our committee in the near future".
"Last November, the American people voted overwhelmingly for Congress to do two things - address the core issues that affect their daily lives, and fulfill our Constitutional responsibility to serve as an independent check and balance on the executive branch by restoring accountability and transparency", Mr Cummings said in a statement.