A lawyer for Paul Manafort, the president’s onetime campaign chairman, repeatedly briefed President Trump’s lawyers on his client’s discussions with federal investigators after Mr. Manafort agreed to cooperate with the special counsel, according to one of Mr. Trump’s lawyers and two other people familiar with the conversations.

The arrangement was highly unusual and inflamed tensions with the special counsel’s office when prosecutors discovered it after Mr. Manafort began cooperating two months ago, the people said. Some legal experts speculated that it was a bid by Mr. Manafort for a presidential pardon even as he worked with the special counsel, Robert S. Mueller III, in hopes of a lighter sentence.

Rudolph W. Giuliani, one of the president’s personal lawyers, acknowledged the arrangement on Tuesday and defended it as a source of valuable insights into the special counsel’s inquiry and where it was headed. Such information could help shape a legal defense strategy, and it also appeared to give Mr. Trump and his legal advisers ammunition in their public relations campaign against Mr. Mueller’s office.