Congressional approval for funds for the Trump administration to spend at the southern border has triggered open warfare between a “squad” of high-profile progressive House Democrats and party leaders they accuse of caving to a White House determined to mistreat migrant children.

The split became brutally evident late last month when Alexandria Ocasio-Cortez of New York, Ilhan Omar of Minnesota, Rashida Tlaib of Michigan and Ayanna Pressley of Massachusetts – together known as “The Squad” – voted against when Pelosi moved to pass a Senate bill that provided $4.59bn in border security funding.

Critics said the bill would not ensure that the money was spent to relieve squalid and overcrowded conditions in holding facilities run by Customs and Border Protection (CBP).

Responding on Twitter, Omar said: “A vote for Mitch McConnell’s border bill is a vote to keep kids in cages and terrorize immigrant communities.”

Tlaib declared: “If you see the Senate bill as an option, then you don’t believe in basic human rights.”

Ocasio-Cortez said: “Hell no. That’s an abdication of power.”

On Saturday, Pelosi responded in a New York Times interview, taking aim at The Squad for voting against “our bill”.

“All these people have their public whatever and their Twitter world,” she said. “But they didn’t have any following. They’re four people and that’s how many votes they got.”

In a tweeted response, Ocasio-Cortez said: “That public ‘whatever’ is called public sentiment. And wielding the power to shift it is how we actually achieve meaningful change in this country.” She also defended her use of social media.

The progressive-moderate split is becoming more evident and bitter.

“If the left doesn’t think I’m left enough, so be it,” Pelosi told the Times.

Referring to her long career and past defeats for progressive policy priorities, the speaker, 79, added: “As I say to these people, come to my basement. I have these signs about single-payer [healthcare] from 30 years ago.

“I understand what they’re saying. But we have a responsibility to get something done, which is different from advocacy. We have to have a solution, not just a Twitter fight.”

The Times columnist Maureen Dowd, who had lunch with Pelosi, wrote that the speaker “pinned the blame for the border bill on [Senate majority leader] Mitch McConnell” and said that after it passed, she received a congratulatory phone call from Donald Trump, who was abroad at the time.

“I actually think if he were here, we might have had a better shot” at getting more of what the Democrats wanted in the bill, Pelosi said, adding: “One thing he understands is the public view of things.”

Such bonhomie is likely to further inflame House progressives. Many younger Democrats accuse older members led by Pelosi of refusing to stand up for principle and of being too afraid of defeat should the party work for impeachment, Medicare for All, a $15 minimum wage, free public college, the Green New Deal or other elements of the progressive agenda.

“Leadership is driven by fear,” Corbin Trent, a spokesman for Ocasio-Cortez and a co-founder of Justice Democrats, told the Washington Post. “They seem to be unable to lead.”

“The greatest threat to mankind is the cowardice of the Democratic party,” Trent added.

Ocasio-Cortez told the Post fear of losing is beside the point for new party members.

“What’s important isn’t just winning but fighting,” she said. “I don’t care about losing in the short term, because we know we’re fighting for the long term.”

On Sunday, Tlaib disputed claims by acting homeland security secretary Kevin McAleenan that the additional funds are a way to get children out of CBP custody. But she soon turned her fire on Pelosi.

“It is very disappointing that the speaker would ever try to diminish our voices in so many ways,” she told ABC’s This Week.

Tlaib called on Pelosi to “honor the fact we are there, that 650,000 people are represented by each and every single one of us” and that “all of us have these experiences that I think have been missing in the halls of Congress. Honor that, respect that, put us at the table. Let’s come up with a solution together.”

She said she would not “support anything that is broken and that dehumanizes people”.

Pelosi, Tlaib said, should “uplift the women, especially the women of color, within your caucus … because I’ll tell you more people like us, more people like me that come out to vote, we win, all of us win.”