In a phone interview, Mr. Cobb said he informed the president weeks ago that he wanted to retire. He said he planned to stay at the White House, likely through the end of the month, to help Mr. Flood transition into the new job.

“It has been an honor to serve the country in this capacity at the White House,” he said. “I wish everybody well moving forward.”

Mr. Flood is expected to take a more adversarial approach to the investigation than Mr. Cobb, who had pushed Mr. Trump to strike a cooperative tone. Mr. Flood initially spoke with the White House last summer about working for the president, but the talks ultimately fell apart because Mr. Flood did not want to deal with Mr. Trump’s longtime New York lawyer, Marc E. Kasowitz, who was overseeing the president’s dealings with the special counsel at the time.