I’m sure there are ways he could be fired. He wasn’t appointed king.
The president may remove a director with adequate cause. President Clinton dismissed Director William Sessions halfway through his term on charges that Sessions misused official resources—such as using FBI aircraft for personal trips.
Congress can also remove an FBI director by impeachment. Article II, Section 4 of the Constitution says that any civil officer can be removed if the Senate convicts him of “treason,bribery, or other high crimes and misdemeanors.” A two-thirds vote is required for conviction. (Mueller, of course, has not been charged with any of these.)
In any situation, however, it is much more likely that the president would ask an FBI director to resign rather than fire him. Clinton dismissed Sessions only after he refused to resign.