WASHINGTON — With health care dominating House action, President Obama was looking for another domestic policy victory Sunday: passage of a rewrite of college aid for the needy.

The legislation, piggybacked on the expedited health care bill, would end a 40-year-old program and its reliance on private lenders. It would authorize the government to originate all assistance loans and would use the savings to increase Pell Grants to students.

In the biggest piece of education legislation since No Child Left Behind nine years ago, the bill would direct more than $40 billion over 10 years into higher education, with $36 billion going toward the popular but financially strapped Pell Grant program. If approved in the House, the Senate would take up the bill next week under the same expedited rules used for health care legislation. That means the Senate could pass the education measure by a simple majority, virtually guaranteeing its success despite qualms from some Democrats and GOP opposition.

The House passed the bill last year, but it failed to get action in the Senate. riding shotgun on the health care bill, the legislation can avoid that obstacle.

Private lenders have conducted an all-out lobbying effort against the bill, arguing it would cost thousands of jobs and unnecessarily put the program in the hands of the government. Under the aid program, financial institutions provide college loans at low interest rates, and the government guarantees the loans in the event of default and subsidizes private lenders when necessary to keep rates low.

directing the government to originate loans, the legislation would see savings totaling $61 billion between now and 2019, according to the nonpartisan Congressional Budget Office.

That money would finance a continuation of the Pell Grants and other higher-education assistance in the bill. But about $19 billion would be used for deficit reduction and to offset expenses in the health care legislation.

The legislation is not as generous as the bill the House passed last year. The government anticipates smaller savings than initially foreseen, and the Pell Grant now faces a $19 billion shortfall. The bill provides $13.5 billion to fill that budget hole.

The bill proposes no increases in Pell Grants over the next two years and a modest increase over the five years that follow. The maximum Pell Grant, which a House-passed bill last year would have raised to $6,900 over 10 years, will now increase only to $5,900. The current maximum grant for the coming school year is $5,500.