A $4.5 billion House bill aimed at providing more funding to migrant families detained at the U.S.-Mexico border is posing a challenge to Democratic House Speaker Nancy Pelosi’s grip on her party, as its liberal faction argue that the bill doesn’t go far enough while moderates worry that pushing for perfection will result in inaction at the border.

Calls for more funding at the border come amid reports that children detained entering the U.S. from Mexico are being held under harsh conditions. Department of Health and Human Services Secretary Alex Azar told Fox News on Monday that the situation at the U.S.-Mexico border is dire. Azar said HHS shelters are at capacity and the budget is not there to increase it unless Congress acts.

Customs and Border Protection Chief Operating Officer John Sanders told The Associated Press that Border Patrol stations are holding 15,000 people – more than three times their maximum capacity.

A $4.5 billion House bill aimed at alleviating circumstances like these is up for a vote Tuesday, but liberal Democrats are calling for provisions to strengthen protections for migrant children, and challenge the Trump administration’s border policies. Democrats met on Capitol Hill with Pelosi late Monday to try and reach a compromise. The meeting reportedly eased some Democratic complaints.

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Asked before the meeting about her concerns that Democrats’ push for perfection might result in inaction at the border, Rep. Alexandria Ocasio-Cortez, D-N.Y., called it “a delicate situation.” Afterward, she appeared to have left the door open saying: “My main goal is to keep kids from dying,” before calling the humanitarian bill a “short-term” measure.”

But others weren’t swayed. Rep. Ilhan Omar, D-Minn., said after the meeting: “We cannot continue to throw money at a dysfunctional system. We are not just asking for simple changes to be made into this bill, but to go back to the drawing board and really address this from a humanitarian issue.”

The White House accused lawmakers in a letter earlier Monday of trying to undermine its efforts at the border, arguing that the House package does provide enough money to toughen border security or funds for Trump’s proposed border wall.

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Congress plans to leave Washington in a few days for a weeklong July 4 recess. While lawmakers don’t want to depart without acting on the legislation for fear of being accused of not responding to humanitarian problems at the border, it seems unlikely that Congress would have time to send a House-Senate compromise to Trump by week’s end.

Fox News' Chad Pergram and The Associated Press contributed to this report.