Moreover, they wrote, Mr. Cohen is a witness of “questionable reliability” and his current lawyer, Lanny J. Davis, has acknowledged that Mr. Cohen himself wrote the lines in question “that formed the basis of his guilty plea for lying to Congress, and not anyone else.” (Mr. Cohen reported to federal prison earlier this month to begin a three-year term.)

It is unclear what evidence the committee may have beyond that shared by Mr. Cohen.

The claim that cooperation should come only on an inquiry with “proper legislative purpose” has been made repeatedly by the Trump administration as it denies House Democrats documents and access to witnesses for nearly a dozen continuing investigations. But Democrats say the House’s legitimate oversight role extends far beyond its legislative responsibilities.

The questions about Mr. Cohen’s false testimony and possible pardons are part of a broader inquiry by the committee into possible attempts to obstruct its investigation of Russian election interference and ties between the Trump campaign and Russia. Raising the possibility of legal exposure for lawyers in the case is certain to further inflame tensions between the president’s team and Democrats who control the House.

The two sides are already at a hostile stalemate over witnesses and documents that the House says it needs to conduct legitimate oversight. Mr. Trump and his allies have accused Democrats of abusing their own congressional powers to humiliate the president.

While it is a crime to obstruct a valid congressional investigation and to conspire to make false statements to Congress, it would most likely be a difficult case to bring against lawyers working in the case. Even if it could prove wrongdoing, the committee has little recourse beyond referring the case to the Justice Department, and the special counsel, Robert S. Mueller III, has already declined to investigate or to charge the lawyers involved.

More likely, given the Trump administration’s across-the-board objection to Democrats’ requests, the line of inquiry could end up in court.