South Carolina Rep. Trey Gowdy Harold (Trey) Watson GowdySunday shows preview: Election integrity dominates as Nov. 3 nears Tim Scott invokes Breonna Taylor, George Floyd in Trump convention speech Sunday shows preview: Republicans gear up for national convention, USPS debate continues in Washington MORE (R) said President Trump Donald John TrumpBubba Wallace to be driver of Michael Jordan, Denny Hamlin NASCAR team Graham: GOP will confirm Trump's Supreme Court nominee before the election Southwest Airlines, unions call for six-month extension of government aid MORE's legal team is doing the president a "disservice" by framing Robert Mueller Robert (Bob) MuellerCNN's Toobin warns McCabe is in 'perilous condition' with emboldened Trump CNN anchor rips Trump over Stone while evoking Clinton-Lynch tarmac meeting The Hill's 12:30 Report: New Hampshire fallout MORE's Russia probe around allegations that Trump's campaign colluded with Russia during the 2016 election.

"The notion that he [Mueller] was hired to only go look for criminality, I think, does a disservice to [the president]," Gowdy said in an interview with CBS News's "The Takeout" podcast.

Gowdy, who chairs the House Oversight and Government Reform Committee, said that Mueller's special counsel investigation extends beyond whether Trump's campaign colluded with Russia to how Russia sought to influence the 2016 election.

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Gowdy told CBS host Major Garrett that Trump's team should understand that the controversial dossier authored by Christopher Steele that makes salacious claims about Trump's ties to Russia was not the driving force behind the federal probe.

"I think his lawyers have done a disservice to him, frankly," Gowdy said. "I think allowing it to be framed as an investigation into collusion and collusion only, conflating the [former FBI Deputy Director] Andy McCabe firing, conflating, frankly, the dossier. This investigation would exist without a dossier. And I've said that for the last 12 months. With or without a dossier, this investigation goes on."

Gowdy went on to urge Trump to continue cooperating with the probe, despite what the White House has admitted is the president's "frustration" at the length of the investigation.

"There's a certain level of frustration that comes from anyone who is, quote, under investigation," Gowdy said. "If you believe you've done nothing wrong, you want the investigation concluded as quickly as possible. It is beyond argument that this has cast a pall over the first part of his presidency."

Speculation has grown over the last few weeks over whether Trump will fire Mueller in an attempt to end the investigation. The White House has repeatedly denied that Trump is considering firing the special counsel.

On Thursday, Trump's lead personal attorney handling the investigation, John Dowd, abruptly resigned, following news that Trump asked Deputy Attorney General Rod Rosenstein Rod RosensteinDOJ kept investigators from completing probe of Trump ties to Russia: report Five takeaways from final Senate Intel Russia report FBI officials hid copies of Russia probe documents fearing Trump interference: book MORE to "bring an end to the alleged Russia collusion investigation."