WASHINGTON -- House Democrats are aggressively urging members to oppose a GOP-authored bill to address the border crisis. And with Republican senators voicing opposition of their own -- both to the House bill and to a Senate Democratic alternative -- it appears increasingly unlikely that Congress will send a border bill to the president's desk before leaving for the August recess.

The Senate held its first vote Wednesday on a bill to provide $2.7 billion to deal with the crisis of more than 57,500 unaccompanied minors who have crossed the border illegally since October. The measure passed by a vote of 63 to 33. But aides in the chamber cautioned that they aren't confident it will be able to clear a second filibuster that could prevent passage.

The House, meanwhile, plans to vote Thursday on a package that provides only $659 million in funding, tied to a number of provisions that most Democrats oppose.

House Democratic aides said leadership is whipping members "hard" to oppose the bill. One aide, speaking on background to discuss strategy, said legislative staff was in "late last night on this" urging members to cast "no" votes.

"We’re still in the process of talking to members, but it won’t be many [who vote for the bill]," a Democratic leadership aide said.

House Minority Whip Steny Hoyer (D-Md.) said Wednesday morning that he expects most Democrats to oppose the bill.

"There may be some; we don't think it'll be a large number," he said, according to The Hill.

Democrats have been particularly critical of some of the bill's measures, such as sending National Guard troops to the border and changing a 2008 law so the minors can be deported more quickly, with less time to make a case for why they should be allowed relief.

"We must have a heart, and look into our souls to guide us in our treatment of these desperate children," House Minority Leader Nancy Pelosi (D-Calif.) said Tuesday in a statement. "While we are reminded of the critical importance of passing comprehensive immigration reform, we must do so much more than the Republicans’ unjust and inhumane proposal."

Democratic votes are especially important if immigration hawks in the House Republican conference decide to defect on the bill. Some GOP members said the bill doesn't go far enough, arguing that it should be coupled with a vote to end the Deferred Action for Childhood Arrivals policy that allows undocumented young people who have been in the U.S. for years to apply for temporary relief.

Senate Republicans Ted Cruz of Texas and Jeff Sessions of Alabama criticized the House bill on Tuesday.

"That the House leaders' border package includes no language on executive actions is surrender to a lawless president," Sessions said in a statement. "And it is a submission to the subordination of congressional power."

At a Wednesday evening meeting at his office, Cruz plans to urge House Republicans in person to vote no, the Washington Post's Robert Costa reported.

Heritage Action, an outside group with influence on tea party Republicans, also opposes the bill because it does not block funding for the Deferred Action for Childhood Arrivals policy.

Senate Majority Leader Harry Reid's (D-Nev.) Tuesday suggestion that comprehensive immigration reform could be attached to the bill could further spook Republicans already wary of voting for it.

House Speaker John Boehner (R-Ohio) said Tuesday that he wants to pass a bill before the August recess, but passage is still not a certainty.

"I think there's sufficient support in the House to move this bill," he said. "We have a little more work to do, though."

The border bill will get at least one Democratic vote. Rep. Henry Cuellar (D-Texas) plans to support the legislation, which includes measures he proposed in a bill co-authored with Sen. John Cornyn (R-Texas). That bill was largely criticized by fellow Democrats, but Reps. Ron Barber (D-Ariz.) and Collin Peterson (D-Minn.) co-sponsored Cuellar's bill.

While Boehner faces significant hurdles in passing a border bill, Senate Democrats do too.

A vote to move forward with their bill, introduced by Appropriations Committee Chairwoman Barbara Mikulski (D-Md.), passed on Wednesday morning, meaning they can proceed to debate on the legislation. But several Democrats -- including Sens. Mary Landrieu (D-La.) and Kay Hagan (D-N.C.) -- voted against moving forward.

The bill must get 60 votes in order to end that debate and amendment process, and as of Wednesday morning, a Senate Democratic leadership aide said they do not expect to hit the 60-vote threshold.

"I don’t see a path to 60 on the back end," the aide said.

Were neither chamber to pass legislation before the August recess, it would represent a remarkable show of political futility -- though the failure would be shared, leaving neither side open to a clean political attack.

UPDATE: 3:40 p.m. -- The White House issued a formal veto threat Wednesday afternoon on the House Republicans' funding bill for the border crisis:

Republicans have had more than a year to comprehensively fix the Nation's broken immigration system, but instead of working toward a real, lasting solution, Republicans released patchwork legislation that will only put more arbitrary and unrealistic demands on an already broken system. H.R. 5230 could make the situation worse, not better. By setting arbitrary timelines for the processing of cases, this bill could create backlogs that could ultimately shift resources away from priority public safety goals, like deporting known criminals. This bill will undercut due process for vulnerable children which could result in their removal to life threatening situations in foreign countries. In addition, the limited resources provided in H.R. 5230 are not designated as emergency, but rather come at the expense of other Government functions.