House Democrats complained that Mr. Bolton was stiff-arming them, but said they would not subpoena him because they did not want to get dragged into lengthy court proceedings. Instead, Democrats have suggested that they may cite the refusal to testify as evidence of obstruction of Congress by the president, which could form its own article of impeachment.

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In his letter, Mr. Bolton’s lawyer argued that his client would be willing to talk to investigators, but only if a court rules that he should ignore White House objections.

Mr. Bolton’s former deputy, Charles M. Kupperman, represented by the same lawyer, filed a lawsuit seeking guidance from the federal courts on competing demands by the executive branch, which has ordered them not to testify, and the legislative branch, which has issued subpoenas directing them to. Late Friday, Mr. Mulvaney filed to join Mr. Kupperman’s lawsuit to have a court settle whether the men are obligated to comply with testimony requests.

But Democrats have said they will not pursue time-consuming litigation on the matter, arguing that it is a stalling tactic.

Mr. Mulvaney defied a House subpoena on Friday morning, citing instructions that he was absolutely immune from testifying.

Investigators were already eager to question Mr. Mulvaney after his admission in the White House briefing room last month of a quid pro quo linking the military aid for Ukraine to that country’s announcement of investigations that Mr. Trump wanted. Hours later, Mr. Mulvaney said he was misunderstood and retracted the statement.

At the White House, Mr. Trump derided the coming public hearings as a “hoax,” even as he sought to project confidence that he would not be brought down by a recent stream of damning testimony from officials in his own administration who have cooperated with the impeachment inquiry.