The Justice Department is barred from discriminating against employees or applicants on the basis of political affiliation.
In a second tweet Tuesday, Trump said the special counsel's team should instead be investigating the campaign of his Democratic opponent, whom he described a "totally Crooked Hillary Clinton".
Coons said Gowdy "is stepping forward and putting to rest the baseless allegations by President Trump".
Quoting Rep. Trey Gowdy, R-South Carolina, who noted on CBS earlier Wednesday that Trump "could have picked somebody else" for the position, Trump tweeted, "I wish I did!"
Asked why Trump persists with his "Spygate" claim, Gowdy - who's not running for re-election for his SC seat and has made a point of keeping his distance from the White House on the Russian Federation probe - said, "I've never met or talked to the president".More news: CIA: North Korea Has No Plans to Denuclearize
Trump has repeatedly pointed to, and at times embellished, reports that a longtime USA government informant approached members of his 2016 campaign during the presidential election in a possible bid to glean intelligence on Russian efforts to sway the election.
"If I were his lawyer, and I never will be, I would tell him to rely on his lawyers and his (communications) folks", he said. "And in fact, when I chose to just do it, I said to myself, I said, 'You know, this Russian Federation thing with Trump and Russian Federation is a made-up story, '" Mr. Trump said.
Gowdy pointed out that no one knows whether Mueller is conducting a criminal investigation.
Trump did not reference Gowdy's comments that there was no evidence of the Federal Bureau of Investigation planting a spy in his campaign. Trump asked a crowd at a campaign rally in Tennessee.
"The 13 Angry Democrats (plus people who worked 8 years for Obama) working on the rigged Russia Witch Hunt, will be MEDDLING with the mid-term elections, especially now that Republicans (stay tough!) are taking the lead in Polls", Trump tweeted.More news: Zinedine Zidane Resigns As Real Madrid Coach After Unprecedented Success
Napolitano's reluctance to back Trump's "Spygate" claims was surprising in part because of the Napolitano's previous tendency to peddle conspiracy theories with no evidence. Former White House chief of staff Reince Priebus had previously been told by Mr. Trump in July that Sessions had to go, but Priebus delayed, and Mr. Trump ultimately let go of the idea of forcing Session to resign. At the time, Sessions said he did not know of a specific reason as to why he should do so.
"If he were my client", Gowdy said, "I would say, if you've done nothing wrong, then you need to sit down with Mueller".
Gowdy told Fox News that he thought Trump should grant an interview to Mueller and his investigators in order to clear all this up.
In the latest flare-up, Gowdy and other House Republicans demanded to see classified information about the origin of the investigation into Trump's campaign, including details about the FBI's use of an informant.More news: Two journalists killed as storm drenches US South