Senior House and Senate lawmakers expressed confidence Friday that they were back on track with a financial bailout plan and would be able to hammer out an agreement this weekend, though the timing still is up in the air.

Senate Majority Leader Harry Reid said he hoped to have a proposal ready for a vote early Monday, possibly before markets open — which would mark a turnaround from what appeared to be a collapse of negotiations Thursday evening.

House Speaker Nancy Pelosi said Friday that "progress is being made," although she didn't divulge details at a late-afternoon news conference in the Capitol.

"We will not leave until legislation is passed that will be signed by the president, and we will be working through the weekend to achieve that end," she said.

But the deal to stave off an economic crisis is not done yet, and House Republicans continue to push for an alternative plan that they say would reduce taxpayer costs and win over wavering Republicans to a compromise bailout that contains elements supported by the administration and senators.

The speed of the negotiations could give Democrats cover that the huge government buyout of bad debt isn't on their shoulders alone. It also could allow Republicans to say that they did due diligence to protect taxpayers.

With House Minority Whip Roy Blunt now in charge of negotiating for Republicans, several lawmakers say they feel more optimistic that a deal can be reached.

Blunt said House Republicans think they can build on a framework laid out early Thursday afternoon among a close-knit group of bipartisan negotiators. That deal fell through after Rep. Spencer Bachus, R-Ala., the ranking member on the House Financial Services Committee, said he didn't have the authority to negotiate for Republicans.

In response to the new optimism, Pelosi earlier said it was "a shame that Republicans boycotted our negotiations" of Thursday. She said senior Democrats Sen. Chris Dodd and Frank had been orchestrating a deal for several days, and the GOP put Bachus in the mix without allowing him to represent them.

A senior Senate Republican nearly concurred.

"Yesterday almost had to happen to get House Republicans to wake up and see this is serious," the Republican told FOX News.

On Thursday, President Bush hosted a historic meeting at the White House on the financial crisis. It included not only Hill leaders but presidential nominees John McCain and Barack Obama, as well. But the meeting was divisive, and all sides departed leaving in limbo a $700 billion package to help Wall Street deal with its load of bad mortgage securities.

On Friday, Bush kick-started negotiations with a statement that Congress will pass a rescue plan.

"There is no disagreement that something substantial must be done. The legislative process is sometimes not very pretty, but we are going to get a package passed," he said in a statement aimed at calming financial markets and American citizens. "We will rise to the occasion."

Bush spoke as markets opened more than 100 points down upon an announcement by federal officials that JP Morgan would be buying up Washington Mutual after it suffered the largest failure in U.S. banking history.

Bachus on Friday was removed from the negotiating table and replaced by Blunt, who said Republicans needed to see that safeguards are in place to protect taxpayers. He said a lot of miscommunication had hampered the negotiations.

But he added that the essence of the deal has to be about liquidity returning to the market. And he didn't rush to a midnight deadline for a compromise.

"We want to get this right. I am going to the negotiating table. ... It's more important we get this done right than we get it done quickly. There's no reason to impose an artificial deadline here," Blunt said, saying he'd like to have a deal before markets open on Monday, but he isn't going to press it.

While many aspects of the deal discussed Thursday remained in play — no "golden parachutes" or severence packages for executives who helped companies fail; congressional oversight that includes transparency and accountability; and a way to get taxpayer money back — House Republicans were still floating an alternative plan.

According to Republican Rep. John Campbell, a member of the GOP negotiating team, the Republican plan would require a new government entity, modeled after the Federal Deposit Insurance Corp., that charges premiums to insure risky mortgages.

The more toxic the mortgage, the higher the premium, Campbell said, but the mortgages would then have government-backed guarantees against future losses, which in theory would draw private investors to purchase it from the institutions and get new liquidity into the mortgage-backed securities market. This would be paid by the mortgage industry, not taxpayer dollars, according to the GOP vision.

Any instutition that participated would have to suspend dividend payments temporarily to free up funds to pay the insurance premiums, which Campbell acknowledged could be steep. If necessary, institutions could borrow from the Federal Reserve to pay the premiums.

The Republican alternative would also provide new capital by allowing recent operating losses to be carried back for up to 10 years and to provide immediate, giant tax refunds to all the institutions. However, Republicans acknowledge that a capital gains tax cut proposal has been dropped due to cost factors and Democratic pushback.

Republicans are estimating the package to be somewhere between $100 billion and $500 billion, less than the $700 billion proposed. If a private/public sector solution is at hand, Republicans will get a mechanism they prefer and the political cover they need to vote for the finished package.

But Reid has already said that he thought the insurance idea was a no-go.

"Secretary Paulson has said their insurance programs won't work. Whether he will say that publicly, I don't know," he said.

White House spokeswoman Dana Perino did not include GOP details in negotiations being considered by the administration, and she said the "insurance suggestion" is unlikely to "take over the whole program."

"One of the things that understandably people wanted to have was more transparency and oversight. So we are willing to work with them on that," she said. "In addition to that, there's the issue of executive compensation ... and Hank Paulson is trying to work with his team on the details as to what that would look like."

But Blunt and other Republicans say the GOP package needs to get an airing.

"Do you want to negotiate in a way that is truly bipartisan ... or do you want a deal that every Democrat is going to be happy about?" Blunt asked.

FOX Business Network's Peter Barnes, FOX News' Chad Pergram and Trish Turner and The Associated Press contributed to this report.