A member of Fuse Washington paid to attend a Republican fundraiser last week where Mr Nunes made his remarks.
"If Sessions won't unrecuse and Mueller won't clear the president, we're the only ones, which is really the danger", Nunes (R-Calif.) said, according to the tape.
Devin Nunes, the House Intelligence Committee chairman and one of President Donald Trump's most vocal supporters in Congress, is anxious about the Republican Party's prospects in the November midterm elections. "The duty of Congress is to be a check and balance on the Executive Branch, and to pursue the facts wherever they may lead", Rep. Ted Lieu, D-Calif., tweeted. He called this a "classic Catch-22 situation", appearing to confuse a "tough spot" - also his words - with a situation in which contradictory conditions make escape impossible.
Nunes, an ally of Trump's who helped on the president's transition team, recused himself from presiding over the investigation after questions arose regarding his handling of classified information. "It's a matter of, it's a matter of timing".
"If we actually vote to impeach, what that does is that triggers the Senate then has to take it up", Nunes said on the audio, adding, that the "Senate only has so much time".
McMorris Rodgers can be heard briefly interjecting into Nunes' discussion of potential impeachment proceedings, but hasn't said publicly that she would support removing Rosenstein.
The remarks appear to be a blunt assessment from Nunes, a top Trump ally in Congress who Democrats accuse of trying to help the White House with the Russian Federation probe, that Republicans must keep their House majority in the midterm elections to shield Trump from the special counsel investigation. "We do not keep the majority, all of this goes away".
Democratic lawmakers and pundits quickly condemned the remarks, saying they indicated Nunes was attempting to obstruct justice by suggesting Rosenstein's investigation, headed by special counsel Robert Mueller, should be quashed.
AUDIENCE MEMBER: "But also, on things that came up in the House on Rosenstein impeachment thing". Okay, now we have a problem, right? "Well, if that's the case, then that's criminal". "I don't think we should be cavalier with this process or term". Since Jeff Sessions is recused on the matter of the Russian Federation probe, the responsibility of deciding whether or not to fire Mueller, if Trump were to ask for this, falls to Rosenstein.
Nunes also said that an effort by some hard-line House Republicans to impeach Deputy Attorney General Rod Rosenstein is on hold because of the election and because an impeachment debate would tie up the Senate and prevent it from quickly confirming Trump's Supreme Court nominee, Brett Kavanaugh. And then take the risk of not getting Kavanaugh confirmed.