The government's plan, proposed by Henry Paulson, the US treasury secretary, had been under discussion in congress where legislators have called for changes.
US Republicans proposed an alternative bail-out plan, but the move was criticised for stalling government efforts to stay the country's economic meltdown.
Talks on the government's bail-out plan, which aims to use taxpayers' money to buy up bad debt held by US financial institutions, began again on Friday.
Agreeing with Bush, Harry Reid, the Democratic senate majority leader, said: "We're going to get this done and stay in session as long as it takes to get it done. We need the House Republicans to join us".
Asian and European stock markets closed down on Friday as the bail-out plans stalled.
Bush appears to be open to Democratic demands to broaden the package to hard-pressed mortgage holders.
A compromise that would give legislators oversight of the deal's implementation, and US taxpayers an equity stake in a bailed-out company, as well as cap the pay of executives of rescued firms had seemed in the works until the White House meeting.
Inside the White House session on Thursday, John Boehner, the House Republican leader, announced his concerns about the emerging plan and asked that the Republican alternative be considered, aides said.
At an impromptu news conference, Barney Frank, the Democratic chairman of the House of Representatives financial services committee, brandished a single sheet of paper that he said was the sum of new Republican proposals introduced at the last minute after a week of sleepless nights in congress.
Reflecting the unease of many Republicans at the government's unprecedented intrusion into private enterprise, the new plan calls for an independent entity to dispose of bad assets, and a cut in capital gains tax.
Few Americans support the government's proposed bail-out.
"It needs to be modified so that it helps the everyday person who is losing their home," Carole Bailey from California told Al Jazeera.
James Berard, a voter in New York, said: "They should let the companies and the executives suffer, I myself have lost $150,000 from my retirement accounts."
McCain has come under fire for bringing presidential race politics into the debate.
Christopher Dodd, the senate banking committee chairman, called the White House meeting, convened by Bush at McCain's request, a "photo op and political theatre that had nothing to do with us getting to work".
But McCain said legislators were on track for a deal and Steve Schmidt, one of his senior advisers, said it was Reid, who had said McCain's help was needed to help gather Republican support for the plan.
Rob Reynolds, Al Jazeera's senior Washington correspondent, said it did appear as if the injection of presidential politics had had an adverse effect on the delicate process of negotiations to come up with a deal.