ATLANTA — John R. Bolton, President Trump’s former national security adviser, knows about “many relevant meetings and conversations” connected to the Ukraine pressure campaign that House impeachment investigators have not yet been informed about, his lawyer told lawmakers on Friday.

The lawyer, Charles J. Cooper, made that tantalizing point in a letter to the chief House lawyer in response to House committee chairmen who have sought Mr. Bolton’s testimony in their impeachment proceedings, arguing that his client would be willing to talk but only if a court rules that he should ignore White House objections.

Mr. Cooper did not elaborate on what meetings and conversations he was referring to, leaving it to House Democrats to guess at what Mr. Bolton might know. Mr. Bolton has been one of the most anticipated witnesses because other current and former officials have described him as deeply disturbed by the effort to pressure Ukraine to provide incriminating information about Democrats to help the president’s domestic political prospects.

Mr. Bolton did not show up for a deposition scheduled on Thursday because, his lawyer said, he wants a judge to determine whether he and his former deputy, Charles M. Kupperman, should testify in defiance of the White House. In effect, Mr. Bolton and Mr. Kupperman are asking for a court ruling on competing demands by the executive branch, which does not want them to testify, and the legislative branch, which does.