Democrats would need 50 senators plus a tie-breaking vote from the vice president to alter the filibuster.
The $1.9 trillion stimulus package proposes $400 unemployment checks which Sen. Joe Manchin wants to cut down to $300.
The outburst came as Senators Warren and Sanders stressed the necessity of scrapping the 60-vote threshold.
When the Senate parliamentarian ruled against GOP tax cuts in 2001, Republicans fired him and passed the cuts anyway.
If Senate Democrats refuse to eliminate the filibuster, they will effectively need 60 votes to pass legislation.
Voting rights advocates are calling on Democrats to abolish the filibuster in order to pass pro-democracy reforms.
Last month, they launched an ad campaign against Manchin for his opposition to $2,000 relief checks.
The recent attack on the Capitol by Trump militants has brought renewed interest to the issue.
Rep. Ilhan Omar blasted McConnell for his hypocrisy: “You lost all credibility when you stole a Supreme Court seat.”
Senators Joe Manchin and Kyrsten Sinema have come out against filibuster reform, protecting a key tool for McConnell.