The comments from Alabama Rep. Mo Brooks, who is challenging the appointed incumbent holding Sessions' seat, Sen. Luther Strange, are a further effort to pull the race to the right and oppose McConnell, who has poured money into supporting the Strange campaign through a closely aligned super PAC.
"Inside the Republican conference, Mitch McConnell's got to go," Brooks said at an immigration policy breakfast with reporters hosted by the conservative Heritage Foundation. "Absolutely, he is the head of the swamp in the United States Senate."
The Republican primary for the seat has become a proxy battle over the direction of the GOP. Brooks is one of three strong contenders for the nomination, including Strange and former Alabama Supreme Court Chief Justice Roy Moore. The campaigns believe the three candidates are the likely top vote-getters in the primary, which will advance to a top-two runoff if no single candidate emerges with more than 50% of the vote.
The primary is August 15 and a runoff would be September 26.
The campaign has become a race to the right.
Sessions, who held the seat for decades, was a strong conservative and advocate of hard-line immigration policies before becoming President Donald Trump's earliest Senate supporter and attorney general.
Brooks has sought to cast himself as the staunchest conservative in the race and natural fighter for Trump's campaign promises -- and Wednesday he cast warning shots to McConnell and Trump about following through on their principles.
"This primary, to sum it up succinctly, everybody in this room is familiar with Donald Trump, candidate, pledging that as President Trump he would drain the swamp," Brooks said. "In this Senate race, the swamp is fighting back and the swamp's candidate is Luther Strange. ... All those swamp critters feels like he's at home and he is at home."
Brooks wouldn't say whom he would back over McConnell, saying there are "a number of other senators" he would support.
While Brooks didn't directly answer whether he would back a primary challenger to McConnell, he did suggest he would want him out of the Senate entirely.
"I want to see -- out of the Senate and out of the House -- people who see the United States Congress as a toy or a plaything or as a trophy or as a thing of stature," Brooks said, saying he wants to support lawmakers who would tackle the US debt that he said he is sure will cause the insolvency of America.
Brooks is a known headache for Republican leadership in the House, where as a member of the House Freedom Caucus he will often defy the party leaders in holding a strong line on government spending.
Brooks also noted multiple complaints with McConnell, including that he supports the 60-vote legislative filibuster that prevents any substantive legislation from getting passed without "groveling at the feet of (Minority Leader) Chuck Schumer."
"He's wrong and it's hurting America," Brooks said. "I would do what I can with my one vote (if elected) to go back to majority rule."
But he also noted the personal feelings involved, citing attack ads that have been run against him supporting Luther Strange by McConnell's PAC that he said painted him as, he said, a liberal.
"If you had seen some of the ads that McConnell has run ... you might understand my consternation," Brooks said. "The deceit coming from the Senate leadership is remarkable, and that's coming from a person who has been active as a candidate ... for 35 years."
Warnings for Trump
Brooks also sent a warning to Trump about staying true to his promises and attacking Sessions, saying pushing out the attorney general would give his voters "mixed feelings."
"A lot of people who are conservatives have great respect for Jeff Sessions and what he has accomplished," Brooks said. "The respect for Jeff Sessions is so great that I'm sure if President Trump were to fire Jeff Sessions or embarrass Jeff Sessions into quitting, that a lot of President Trump voters, particularly in the primary, will have mixed feelings about what is going on," Brooks said.
Brooks said he fears Trump is making efforts to push Sessions out, saying Trump has "started disparaging" Sessions and that is "very counter-productive."
"It reminds me of the kinds of public statements we saw before Comey was terminated," Brooks said. "I believe the President should not be making the kinds of personal and other derogatory statements about Jeff Sessions, who I know to be a very fine man, a very honorable man and a very diligent public servant. I believe those statements publicly are very counter-productive. I am concerned there is increasing pressure by the president to cause Jeff Sessions to quit."
Having made border security and tough immigration enforcement a centerpiece of his campaign, Brooks said he would be concerned about the future of those issues under the Trump administration without Sessions.
"If he is ousted by President Trump, that causes me to have great concern about where our next attorney general will take our illegal alien control efforts," Brooks said.
Brooks also put Trump on notice to stay true to his campaign promises.
"With the respect to the policies that Donald Trump professed to believe in during the campaign, I think you would find me, Mo Brooks, as a United States senator agreeing with those policies the vast majority of the time, perhaps even more so than President Trump does," Brooks said. "There's a difference between candidate Trump and President Trump."
He cited Trump's promise to get Mexico to pay for the border wall.
"It's been six months into his administration and I've yet to see it," he said of proposals that would accomplish that.
Brooks also said he was disappointed Trump didn't veto a government funding bill in April that did not include border security funding.