Washington (CNN) The House approved Wednesday an initial $7.9 billion package of disaster relief funds in the wake of Hurricane Harvey, while Senate leaders appear likely to attach the bill to a politically-fraught effort to raise the debt ceiling.

The vote was 419-3. The bill moves to the Senate, which is expected to vote on the plan this week, the bulk of which will go to Federal Emergency Management Agency, which has warned that it is running out of resources rapidly. In addition to supporting the recovery efforts in Texas, the agency is preparing for Hurricane Irma, another powerful storm that is approaching Puerto Rico and threatens to have a devastating impact on Florida.

But in a new wrinkle that could complicate the debt ceiling fight, Democratic leaders, including House Minority Leader Nancy Pelosi and Senate Minority Leader Chuck Schumer, released a statement Wednesday morning backing only a three-month debt limit increase as part of the Harvey aid package.

While Republican leaders haven't said how long they plan to extend the debt limit, it was expected that it would be longer than three months.

"Democrats are prepared to offer our votes for the Harvey aid package, and a short term debt limit increase of three months," Schumer and Pelosi said. "Given Republican difficulty in finding the votes for their plan, we believe this proposal offers a bipartisan path forward to ensure prompt delivery of Harvey aid as well as avoiding a default, while both sides work together to address government funding, Dreamers, and health care."

House Speaker Paul Ryan called the Democratic leaders' push for a short, three-month increase in the debt limit "a ridiculous idea."

"I think that's ridiculous and disgraceful that they want to play politics with the debt ceiling at this moment when we have fellow citizens in need to respond to these hurricanes so that we do not strand them," Ryan told reporters.

For now, the bill that the House will vote on Wednesday -- which does not include a debt ceiling attachment -- is predicted to pass easily, according to Rep. Steny Hoyer, the number two House Democrat. He said Tuesday that he told House Majority Leader Kevin McCarthy "almost every Democrat, if not every Democrat, would support it and I expect it to pass with overwhelming bipartisan support."

McCarthy warned members the House would remain in town to complete work on Harvey aid and depending on how the Senate votes that could mean a final House vote late Friday or Saturday -- according to multiple House GOP members.

Last week, the White House requested $7.85 billion for response and recovery efforts. Most of that money would go to the Federal Emergency Management Agency, while about $450 million would go to the Small Business Administration's disaster loan program.

Congress is also expected to authorize another $6.7 billion in expedited aid relief that will be part of a stopgap spending bill to keep the government open until the end of the year. The funding measure must be approved by this month's end to avoid a government shutdown.

These numbers, which combined amount to about $15 billion, are just for immediate spending needs after Harvey. Texas Gov. Greg Abbott said Sunday on CNN's "State of the Union" that he believes the federal government will ultimately need to provide far more than $100 billion. And multiple Republicans tell CNN that with another hurricane threatening to impact Florida later this week and more detailed assessments coming in terms of the costs of recovery is Texas they expect more votes on additional disaster relief this fall.

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With a pressing deadline at the end of the month to raise the debt ceiling or go into default, Treasury Secretary Steven Mnuchin said the White House wants Congress to attach the debt ceiling bill to disaster relief aid. Such a strategy might make it easier for the debt ceiling bill to pass, since Republicans are divided on raising the debt limit but Harvey aid has wide bipartisan support.

House Republican leaders recognize that a majority of their own members don't want to vote for a measure to raise the debt ceiling. They expect that the Senate -- after the House votes Wednesday -- will take the Harvey funding bill and tack on a provision to avoid a default, and when it is sent over to the House it could pass with mostly Democratic votes.

"I'm willing to not say that hurricane relief has to have offsets," said Rep. Mark Meadows, chairman of the conservative House Freedom Caucus. "I'm willing to do a number of things that a typical conservative would not support because I see the devastation that's there, but yet at the same time tying a debt ceiling to it is just using it as leverage to get something done that perhaps should have gotten done in a different manner."

Meadows is referencing how conservatives in the House are not pressing so far for the Harvey aid to be accompanied by any spending cuts or offsets -- a position many of them took in 2013 when Congress voted on emergency money after Superstorm Sandy. But several leading Republicans on the right of the House GOP conference are loudly complaining about the expected strategy to tie the increase to the debt limit to the disaster relief legislation.

"I think it's a terrible idea," Meadows said. "Anytime you use a tragedy to advance something that should have had a plan without a hurricane happening is not an appropriate approach."

North Carolina Rep. Mark Walker, the leader of another group of fiscal conservatives, said in a written statement, "If we resort to just kicking the can down the road on the debt, it only shows that Republicans do not take the problem of our $20 trillion debt seriously. Republicans passed spending reforms with Barack Obama as president. We have no excuse why we can't do it now with Republican leadership."

On the Senate side, Majority Leader Mitch McConnell said Tuesday the three most important items on the GOP's agenda are passing disaster relief, preventing a default, and funding the government -- though he did not provide specifics for how and when they will do these priorities.

Sen. John Cornyn, the number two Republican in the Senate, said he would be "open" to a bill that attaches a debt ceiling increase to Harvey aid.

"I would support that," Cornyn told CNN as he arrived back at the Capitol from Texas, where he has been dealing with hurricane relief and clean-up efforts.

While his vocal support suggests the idea could get broad support from Republicans, Cornyn, who is the GOP whip, said he had not had time to gauge all his colleagues -- though he added he "believe(s) that's the plan."

"I'm told if we don't raise the debt ceiling, then we can't appropriate the additional funds for Harvey on an emergency basis, which we absolutely need to do," he added. "I continue to be worried about the debt, but I don't think this is the time to have that debate."