Many members of Congress, like Mr. Cuellar, had expected Mr. Kelly to be a force for stability in a White House that has at times seemed consumed with dysfunction, but have found a different reality.

“He brought some order to the chaos that was there, but it’s a long way from a functioning White House,” said Senator Richard J. Durbin of Illinois, the second-ranking Democrat, said in an interview for “The Daily,” a New York Times podcast. “His role from time to time I’ve considered to be destructive, sometimes constructive. I wonder, you know, if he really is the man I thought he was when I voted for him as secretary of D.H.S.”

Mr. Durbin, who has clashed with Mr. Trump over his assertion that the president used a vulgar term in an immigration meeting last week, was referring to the Department of Homeland Security, which Mr. Kelly led before taking his current position.

Mr. Kelly has made little secret of the fact that he never wanted to be White House chief of staff, and took the job out of the same sense of duty that led him to a four-decade career in the Marines.

In the West Wing, Mr. Kelly seldom allows the staff to forget the dynamic, according to people who have observed him, often positioning himself as a one-man check against dangerous or reckless moves by the commander in chief. His loyalty is not to the president, “but to the Constitution and the country,” he has said, according to two people with direct knowledge of his remarks.

Mr. Kelly, officials say, has made a conscious decision not to focus as much on curbing the president’s penchant for tweeting or saying inflammatory things, and to instead pour his efforts into controlling who sees and talks to Mr. Trump and trying to shape his thinking on key issues.

“I have said many times I was not put in this job to change the way the president of the United States does business,” he said in an interview with Fox News on Wednesday. “I was put in the job to make sure the staff process better informs him on a range of issues.”