House Speaker Paul Ryan said Tuesday that the special counsel "should be free" to complete his investigation into Russian meddling in the 2016 election and links between the Trump campaign and the Kremlin.

"The special counsel should be free to follow through his investigation to conclusion," Ryan said at a news conference Tuesday morning. The Wisconsin Republican also said he has received assurances that Trump won't fire Robert Mueller. "We have a system based on rule of law," the speaker added.

Ryan's statement comes as Republicans face criticism for not taking a hard stance against the possibility that President Donald Trump could fire the special counsel. The speaker's remarks largely line up with other GOP lawmakers' statements on Mueller.

Senate Majority Whip John Cornyn, a Republican from Texas, has said he doesn't see a need to use legislation to protect Mueller. Sen. Lindsey Graham, R-S.C., has said that firing Mueller would mark the "beginning of the end" of Trump's presidency, although he has also said that he has no concerns about a potential termination.

Trump and one of his lawyers, John Dowd, reignited concerns that the president might fire the special counsel over the weekend, after Attorney General Jeff Sessions fired former FBI Deputy Director Andrew McCabe two days before his retirement. Trump often targeted McCabe in harsh tweets, and lumped in his firing with the Mueller investigation. The president also blasted former FBI Director James Comey in his weekend tweet storm. Trump fired Comey in May.

Sunday night, another Trump lawyer, Ty Cobb, followed up by saying that the president wasn't thinking or talking about firing the special counsel.

Yet on Monday, Trump again spurred speculation that he could terminate Mueller, tweeting, "A total WITCH HUNT with massive conflicts of interest!" The president has often referred to the investigation as a "witch hunt." On Tuesday, Trump declined to answer when a reporter asked whether he wanted Mueller fired.

Trump has repeatedly denied colluding with the Russians.

The legal team representing the president in the Russia probe is also in upheaval, according to various media reports. Dowd told CNBC on Tuesday, however, that the team remains intact.

— CNBC's Kevin Breuninger contributed to this report.