Mr. Giuliani did not consult every member of the president’s legal team, or the network of lawyers around Washington whose clients have been entangled in Mr. Trump’s legal disputes, according to several people close to the team. Emmet T. Flood, a lawyer hired by Mr. Trump on Wednesday, was not involved in Mr. Giuliani’s plans to reveal the payments to Mr. Cohen during an interview with Sean Hannity on Fox News, one of the people said.

The abrupt disclosure — which even caught Mr. Hannity, a confidant of the president’s, by surprise — set off a flurry of calls between Mr. Trump’s lawyers as they sought to determine whether Mr. Giuliani meant to reveal the president’s reimbursement. Witnesses and lawyers around Washington scoured transcripts, watched television clips and called each other in an effort to grasp the consequences of what Mr. Giuliani had said.

The president’s other lawyers ultimately determined that Mr. Giuliani had consulted with Mr. Trump, people close to them said, but were left speechless about why he decided to make the disclosure in such a high-profile way and without any strategy to handle the fallout.