(CNN) House Speaker Nancy Pelosi is working furiously to push through a multibillion-dollar border funding package amid Democratic infighting over whether the leadership is employing the right strategy to confront a major humanitarian crisis at the border.

According to a senior Democratic aide, Pelosi expressed a desire to review those proposals, which would be incorporated into the spending bill that's set for a vote on the floor Tuesday.

Internal division among House Democrats erupted during a meeting Monday night. Pelosi held a late night, two-and-a-half hour meeting in her office that one member described as "very tense" with "high decibel levels."

The meeting included high profile members of the Congressional Progressive Caucus -- like Rep. Rashida Tlaib of Michigan, Rep. Ilhan Omar of Minnesota and Rep. Alexandria Ocasio-Cortez of New York -- as well as some members of the Congressional Hispanic Caucus who have concerns and questions about the bill, mostly over whether it does enough for humanitarian needs.

During the meeting, House Appropriations staff and lawmakers faced a barrage of questions from Democrats angry about the contents of the bill and the leadership's strategy, according to sources who attended.

At the meeting, Pelosi asked the members where they stood on the bill, signaling she could be open to making some last-ditch changes. But she indicated she wants to push the bill through the House as soon as Tuesday, a major gamble given the divisions that still remain in the caucus.

"This is the hardest she's had to work for a bill in her speakership," one Democratic source at the meeting said. "I just think the leadership approached this wrong."

Defending the approach, Pelosi told her colleagues she wanted the House to pass a measure that President Donald Trump would ultimately be forced to sign into law.

In the room with roughly two dozen members, Rep. Jan Schakowsky, of Illinois -- a close Pelosi ally -- gave an impassioned defense of the bill, arguing it must pass the House this week given the money running out for an office within the Department of Health and Human Services that deals with housing unaccompanied minors in federal custody.

But House appropriators faced tough questions from a number of lawmakers, including Ocasio-Cortez and Reps. Jimmy Gomez of California, Barbara Lee of California and Lloyd Doggett of Texas.

Critics said the funding bill would not do enough to protect migrants and would pump money to a broken system without fixing it.

Moreover, Democratic critics fumed that their bill should go much further, given they will have to reconcile the House plan with one emerging from the GOP-led Senate. They argued they would be in a weaker negotiating position with the Senate if the House bill did not go as far they'd like.

Still, Rep. Ben Ray Lujan, the No. 4 Democrat, emerged from the meeting expressing confidence that a vote will still happen Tuesday and that Democrats will have enough support to pass it.

Rep. Veronica Escobar, of Texas, said the two caucuses had conference calls about the bill over the weekend, where questions and confusion arose about what exactly was in it. Escobar said that's why she asked leadership to hold the meeting Monday night.

She said she's planning to vote for the bill and downplayed the notion that any changes could be made, saying the House is "running out of time."

A spirited Ocasio-Cortez entered the meeting frustrated with the "caging of children" and saying she won't vote for any bill that funds existing policies.

"We need to stop funding the detention of children under any and all circumstances," she told reporters.

"I will not fund another dime to allow (Immigration and Customs Enforcement) to continue its manipulative tactics," she later added.

Rep. Tony Cardenas, of California, said it was a matter of making sure people understood the House bill was better than the Senate bill.

"We want to make sure people understand it and not allow perfection to be the enemy of the good."