Rep. Ocasio-Cortez, of New York, and Rep. Omar, of Minnesota, are supporting efforts to have Mr Reed, Mr Biden's former chief of staff, frozen out of the Office of Management and Budget (OMB).
Reports emerged earlier this week suggesting that Mr Reed, who also served in the Obama administration, was top of the president-elect's list to head up the OMB.
But campaigning organisations and some on the left of the Democrat Party - congresswomen Ocasio-Cortez and Omar some of the most high-profile among them - oppose the move.
They say Mr Reed is a "deficit hawk" who presided over social security and Medicare cuts during the Obama years.
Following the news that Mr Reed was the frontrunner to take up the OMB role, Justice Democrats, a political action committee, launched a petition calling on the Biden/Harris transition team to "prioritize working people, not Wall Street deficit scaremongers".
"Joe Biden must not repeat Obama’s mistake," the petition reads. "Rejecting Reed will be a major test for the soul of the Biden presidency."
The petition was also signed by fellow Democrat "squad" member Rep. Rashida Tlaib (Michigan) according to Axios. They are the first sitting members of Congress to sign it, the outlet reported. Incoming representatives Jamaal Bowman and Cori Bush also put their names to the document.
"If the Biden administration is serious about protecting Medicare and social security, they must not appoint one of the biggest champions of cuts to lead their budget agency," Congresswoman Omar said after sharing the petition on Twitter.
Tensions between the two wings of the Democratic party have been simmering in recent weeks following their poor showing in House elections, with each side blaming the other for a loss seat while the Republicans made gains.
The left of the party is calling on Mr Biden to include some of its representatives in his top team, arguing that it was support for more progressive candidates that put him in the White House.
Centrist Democrats believe such moves could put off voters at upcoming elections who were spooked by what they said was the party's "socialist messaging" during the presidential election.
Mr Biden announced his first round of Cabinet picks on Tuesday and it did not include any figures from the left of the party. But the nomination of Anthony Blinken for the secretary of state role went down well.
Faiz Shakir, Bernie Sanders’s 2020 campaign manager, said the section was a "solid one". The nomination of John Kerry to a climate change role and Janet Yellen, Mr Biden's choice for Treasury secretary, were other picks viewed positively.
Elsewhere, Alejandro Mayorkas was announced as Department of Homeland Security secretary and Avril Haines was pickled for director of National Intelligence.
Linda Thomas-Greenfield was chosen as UN ambassador in what vice president-elect Kamala Harris described as a Cabinet that "looks like America".
Mr Biden insists that he is determined to pursue a "progressive" agenda. Last night, he said, "there's nothing really off the table" when asked if he had consulted senators Elizabeth Warren and Bernie Sanders about Cabinet choices.
Those comments came during his first sit-down interview since the election. Speaking to NBC News's Lester Holt, Mr Biden vowed that his administration would not be "a third Obama term".