WASHINGTON — Robert S. Mueller III managed in a dozen years as F.B.I. director to stay above the partisan fray, carefully cultivating a rare reputation for independence and fairness. But his appointment as special counsel overseeing the Trump-Russia investigation has thrown him into the middle of the most charged political brawl of his career — especially since his early hires include several prosecutors who have donated to Democratic candidates in the past.

“You are witnessing the single greatest WITCH HUNT in American political history — led by some very bad and conflicted people!” President Trump wrote on Twitter on Thursday morning. The president’s supporters and conservative news outlets have echoed his message that Mr. Mueller’s investigation is unjustified and its staff biased against Mr. Trump.

Veterans of past Washington battles on the borders of law and politics said that the president’s pushback was to be expected, but that its ferocity and timing were unusual. Just one month into the job, Mr. Mueller has not yet finished hiring staff members or installing a computer network — deliberately segregated from the main Justice Department — in the Patrick Henry Building in downtown Washington.

“It’s early in the game to begin to impugn the prosecutors,” said Philip Allen Lacovara, a Watergate prosecutor and a Republican. “It’s a pre-emptive nuclear strike. If you’re afraid of what the prosecutors are going to find out, you try to debunk anything they might come up with in advance by attacking them.”