People close to the president said Mr. Whitaker first came to the attention of Mr. Trump because he liked watching Mr. Whitaker express skepticism about aspects of Mr. Mueller’s investigation on television.

In August 2017, Mr. Whitaker appeared on CNN and said that if Mr. Mueller began investigating the Trump Organization, “I think that would be crossing the red line.”

In a CNN interview the month before, Mr. Whitaker offered a situation in which Mr. Trump could try to hobble Mr. Mueller’s investigation behind the scenes by pressuring the Justice Department to cut the special counsel’s budget.

He said that situation was “a little more stage-crafty than the blunt instrument of firing the attorney general and trying to replace him.”