The president’s tweets, posted on a Saturday in which he remained inside the White House with no public schedule, came as Mr. Mueller is said to have sent questions to Mr. Trump’s legal team as part of negotiations over an interview with the president. Mr. Mueller is seeking the interview, according to two people close to the White House, in order to ask follow-up questions, but put forward the list as a start.

Mr. Dowd’s remarks about Mr. Mueller’s investigation represented an extraordinary shift in public strategy for the Trump legal team. Since taking over the case last summer, Mr. Trump’s lawyers have urged a strategy of restraint, in which the president avoids discussing Mr. Mueller or criticizing him, and the lawyers had done nothing publicly until now that could agitate the special counsel’s team.

The comments by both Mr. Trump and Mr. Dowd lent credence to Mr. McCabe’s assertion that the president sees his firing as directly tied into Mr. Mueller’s case.