Under pressure from the White House, Senate leaders are quickly moving forward with a plan to change how the government sets federal student loan interest rates, tying them to market rates but imposing caps on how high those rates can go.

Senate Majority Leader Harry Reid (D-Nev.) said Thursday that a vote could come this week, but senators who worked on the proposed legislation said next week is more likely.

A bipartisan group of senators reached a compromise on Thursday morning following weeks of negotiations, as Department of Education staffers camped out in their offices. On Tuesday, the senators ventured to the White House to meet with President Obama, who urged them to make a decision.

Under this new proposal, undergraduates would all pay the same interest rate, a change from recent years when some low- and middle-income students received a lower rate. Graduate students and parents of students would pay higher interest rates with higher caps.

“While this isn’t the agreement any of us would have written -- and many would like to see something quite different -- I believe that we have come a long way in reaching common ground on a very, very difficult and challenging topic,” Majority Whip Richard J. Durbin (D-Ill.) said at a Thursday afternoon media event, standing in front of the seven senators who worked on the legislation.

For the coming school year, undergraduates would see rates of 3.86 percent. That’s lower than the current fixed rate of 6.8 percent, but the new rate could go as high as 8.25 percent in future years.

Graduate students would have a 5.41 percent interest rate for the coming year and up to 9.5 percent in the future. PLUS loans, which can be taken out by parents of students or graduate students, would have an interest rate of 6.41 percent and could go as high as 10.5 percent. Right now, graduate students have interest rates of 6.8, and PLUS loans are at 7.9 percent.

Under this plan, the government is expected to make $715 million over the next decade. Sen. Lamar Alexander (R-Tenn.) said this was not intentional and is instead the result of small gains on hundreds of billions of loaned dollars over 10 years. He said he hopes that it will be closer to zero.

“Our goal is to make it neutral, in terms of fair to taxpayers, fair to students,” Alexander said.

Senate Democratic leaders had originally backed a bill that would extend a low interest rate on one type of student loan, buying them more time to tackle a full reform of all interest rates. That legislation failed in a procedural vote, forcing them to work with Republicans to solve the problem now.

The compromise was based on a bill introduced earlier this year by Joe Manchin III (D-W.V.), Richard Burr (R-N.C.), Tom Coburn (R-Okla.), Angus King (I-Maine), Tom Carper (D-Del.) and Alexander. The final deal was agreed upon by those six senators, along with Durbin and Sen. Tom Harkin (D-Ia.), chairman of the education committee.

The group was close to having a tentative plan last week, but then learned that it would cost the government $22 billion over 10 years — something that Republicans would not back. Working off that framework, the negotiators increased all of the rates at least slightly in an effort to reduce the cost to the government.

A key contention point has been the caps on these loans. Democrats have pushed for low caps that benefit students, even if it costs the government. While negotiators have stayed committed to a cap of 8.25 percent on undergraduate loans, they have allowed the caps on rates for graduate students and parents to increase.

Harkin, in particular, has pushed for lower caps to prevent future borrowers from double-digit and near double-digit rates, a battle that he appears to have lost. But the estimated interest rates for the coming 10 years are all well below the caps, according to projections provided by a Democratic aide.

“What I held out for was getting a good deal for students, and I think in the art of compromise, that’s what we got,” Harkin said. “The thing is, students are going to have a better deal than what they would have had otherwise.”

At the last minute, Harkin added a provision to the legislation that would require a Government Accountability Office study later this year of the cost of running the government’s education loan programs. Harkin said that the outcome of that study could prompt lawmakers to yet again tweak the interest rates.

The House has passed its own student loan bill, which also sets interest rates based on the market. If the Senate passes its bill, differences between the two will have to be reconciled. The House’s plan would allow interest rates to change over the lifetime of a loan, while the Senate proposal would lock those rates in when the loan is taken out.

Asked about the Senate plan on Thursday, House Speaker John Boehner (R-Ohio), said it seemed similar to the measure passed by the House. “So when we see the details, I’m hopeful we’ll be able to put this issue behind us,” he said.

Lawmakers are pushing to get this done before they leave for a long recess in August — the point at which many students will likely sign loans for the coming school year.

Ed O’Keefe contributed to this story.