US President Donald Trump has called his former Secretary of State Rex Tillerson "dumb as a rock" and "lazy as hell" a day after the former diplomat publicly said he had warned the president against doing things that would violate the law.

"Mike Pompeo is doing a great job, I am very proud of him. His predecessor, Rex Tillerson, didn't have the mental capacity needed. He was dumb as a rock and I couldn't get rid of him fast enough. He was lazy as hell. Now it is a whole new ballgame, great spirit at State!" Trump wrote on Twitter.

Trump fired Tillerson on March 13 after a series of public rifts over North Korea, Russia and Iran policy, dismissing the former Exxon Mobil Corp chief executive in a tweet.

In addition to policy disputes, relations were strained by reports that Tillerson privately called Trump a "moron."

In an interview with CBS News political contributor Bob Schieffer on Thursday, Tillerson described Trump as "pretty undisciplined, doesn't like to read, doesn't read briefing reports ... doesn't like to get into the details of a lot of things."

Asked how his relationship with Trump had gone off the rails, Tillerson said it might have reflected what he saw as his obligation to tell the president that he could not do certain things because they were illegal or would violate US treaties.

"I'd have to say to him, well Mr. President, I understand what you want to do, but you can't do it that way. It violates the law, it violates treaty," Tillerson said.

"You know, he got really frustrated."