Mr. Bush has indicated he will sign the legislation, despite previous objections to parts of the bill. Specifically, the administration has criticized a student loan interest-rate cut and a new loan-forgiveness program, among other things.
House Democrats had made the popular interest-rate cut a priority during the run-up to the last election in which they regained control of Congress.
The House voted 292-97 for the student aid bill Friday. Earlier in the day, the Senate approved the measure 79-12.
All the lawmakers who voted against the bill were Republicans.
The boost in financial aid to college students was one of half a dozen domestic priorities Democrats set when they took control of Congress this year. Two others - an increase in the minimum wage and mandatory air and sea cargo inspections - already have become law, and a third, ethics reform, is awaiting Mr. Bush's signature.
"This is an exciting day for parents and students who struggle to put together the financial means to pay for college," said Rep. George Miller, D-Calif., chair of the House education committee.
The bill would increase the maximum Pell grant, which goes to the poorest college students, from $4,310 a year to $5,400 a year by 2012.
It also would cut interest rates on federally-backed student loans to poor and middle-class students from 6.8 percent to 3.4 percent over the next four years.
California Rep. Buck McKeon, the leading Republican on the House education committee, criticized the rate cut. He said students will go back to paying the higher rate in four years or taxpayers will have to foot the bill for the cut to continue.
"What once was a campaign promise has become a trap that will ensnare either students or taxpayers," McKeon said.
Democratic lawmakers say the roughly $20 billion in cuts to banks are aimed at excessive government subsidies to lenders. The subsidies were established to ensure that banks enter and stay in the college loan business.
Banking industry officials have objected to the cuts and have said they could adversely affect services provided to borrowers.
Nearly all of the cuts would go toward making college cheaper, but $750 million would be spent on federal budget deficit reduction. The legislation is part of a must-pass bill needed to meet spending targets in the federal budget.
The bill also sets up a loan-forgiveness program for college graduates who work for 10 years in public service professions, such as teaching or nursing.
It also would cap annual payments for students at a percentage of their income, which lawmakers say would prevent people from having to pay back more than they can afford.
"Today, with this bill, we're sending a message, and that message is that no qualified student will be denied a college education because of cost," said Sen. Edward Kennedy, D-Mass., chairman of the Senate education committee.
Once signed by the president, the legislation will begin taking effect Oct. 1.