The talks continued even as it was announced that Goldman Sachs and Morgan Stanley, the last two major investment banks in the US, had agreed to become bank holding companies.
The move is expected to give them easier access to credit and greater protection from the economic crisis.
Referring to the bailout plan, Bush said: "Working together, I am confident we can enact the legislation necessary to prevent lasting damage to our economy and meet the unique challenge facing us today."
However, concern about the cost and lack of details about the plan dragged down the dollar, government bonds and stock markets in the US and Europe on Monday.
This led to increases in the price of gold and oil.
John Terret, Al Jazeera's correspondent in New York, said: "We were expecting... many more details of this plan. The details are very few and far between and that is worrying the markets and driving them down."
Bush warned against adding unrelated provisions onto the bailout plan after Democrats said that any deal must include provisions to guard against potential abuses.
He said the world was watching to see whether the US could act quickly to shore up markets.
Goldman Sachs and Morgan Stanley will now be subject to more stringent regulation, limiting the amount of money they can borrow relative to the capital they have and significantly reducing the huge profits they are able to make.
|Bush says negotiators have made "good headway" on the $700bn bailout [EPA]
The regulation completes an overhaul of the structure of the banking industry, which had been broken up in the 1930s into commercial and investment banks under rules to restore confidence during the Great Depression.
Terrett said: "It's a real sea change... crucially if they get into trouble again they will easily be able to get a credit life line from the federal authorities.
"Depending on your view, it is either a capitulation of Wall Street's old ways... or a very savvy way of operating, looking to the future."
Lloyd Blankfein, the chairman of Goldman Sachs, said: "We believe that Goldman Sachs, under Federal Reserve supervision, will be regarded as an even more secure institution with an exceptionally clean balance sheet."
He said the decision had been "accelerated by market sentiment".
In recent weeks, the companies have seen their former investment bank rivals Lehman Brothers collapse, while Bear Stearns and Merrill Lynch have been taken over.
"This new bank holding structure will ensure that Morgan Stanley is in the strongest possible position," John Mack, Morgan Stanley chairman, said.
Last week's turmoil on the financial markets triggered warnings that the global financial system was teetering on the brink of collapse as a result of the year-old subprime mortgage crisis.
On Sunday, Henry Paulson, the US treasury secretary, urged congress to act quickly to approve the rescue plan and said he was "aggressively" pushing for other countries to come up with their own bailouts for financial institutions.
It is the US's largest ever interventionist package and will see the US national debt limit raised to $11.3 trillion.
If it is approved, Paulson's decisions will not be reviewable by any court, leading some critics to say this effectively makes the treasury boss a dictator of US financial markets.
Paulson defended his actions, which he acknowledged would cost the US taxpayer billions of dollars, during an interview with the NBC programme Meet the Press on Sunday.
"This is not something that we wanted to do. This was something that was very necessary," he said.
"We did this to protect the taxpayer."