Mr. Scaramucci’s abrupt removal came just 10 days after the wealthy New York financier was brought on to the West Wing staff, a move that convulsed an already chaotic White House and led to the departures of Sean Spicer, the former press secretary, and Reince Priebus, the president’s first chief of staff.

The decision to remove Mr. Scaramucci, who had boasted about reporting directly to the president not the chief of staff, John F. Kelly, came at Mr. Kelly’s request, the people said. Mr. Kelly made clear to members of the White House staff at a meeting Monday morning that he is in charge.