WASHINGTON — Senators are ready to offer students a better deal on their college loans this fall, but future classes could see higher interest rates.

The Senate could vote as early as Thursday on a bipartisan compromise that heads off a costly increase for returning students.

“We should get it done as quickly as possible. We can do it today,” Senate Majority Leader Harry Reid said Thursday.

The compromise could be a good deal for students through the 2015 academic year, but then interest rates are expected to climb above where they were when students left campus in the spring.

Under the deal, all undergraduates this fall could borrow at 3.85 percent interest rates. Graduate students would have access to loans at 5.4 percent, and parents would be able to borrow at 6.4 percent. Those rates would climb as the economy improves and it becomes more expensive for the government to borrow money.

The deal was described by Republican and Democratic aides who insisted on anonymity because they were not allowed to be publicly identified discussing the ongoing negotiations.

Undergraduates last year borrowed at 3.4 percent or 6.8 percent, depending on their financial need. Graduate students had access to federal loans at 6.8 percent and parents borrowed at 7.9 percent.

The interest rates would be linked to financial markets, but Democrats won a protection for students that rates would never climb higher than 8.25 percent for undergraduates. Graduate students would not pay rates higher than 9.5 percent, and parents’ rates would top out at 10.5 percent.

The bipartisan agreement is expected to be the final in a string of efforts that have emerged from near-constant work to undo a rate hike that took hold for subsidized Stafford loans on July 1. Rates for new subsidized Stafford loans doubled from 3.4 percent to 6.8 percent, adding roughly $2,600 to students’ education costs.

Lawmakers from both parties called the increase senseless but differed on how they thought the lower rates should be restored.

Democrats had urged a one- or two-year extension of the current rates, while lawmakers plan to do a complete rewrite of the Higher Education Act in the fall to address the larger issue of college costs. Reid said he would allow members of his caucus one more attempt at the tactic but urged his colleagues to back the long-term proposal.

Republicans have pushed for a link between interest rates and the financial markets. Obama included that link in his budget proposal, as did House Republicans. Democrats balked, saying it could produce government profits on the backs of borrowers if rates continued to climb.

Leaders from both parties, however, recognized the potential to be blamed for the added costs in the 2014 elections if nothing were done.

Senate aides said a vote on the agreement could come as early as Thursday, although it could be pushed back to the middle of next week.

The House has already passed student loan legislation that also links interest rates to the 10-year Treasury note. The differences between the Senate and House versions are expected to be resolved before students return to campus this fall, and Obama is expected to sign the bill.

Few students had borrowed for fall classes. Students typically do not take out loans until just before they return to campus, and lawmakers have until the August recess to restore the lower rates. The students who had borrowed for summer programs since July 1 would have their rates retroactively reduced.

The deal was estimated to reduce the deficit by $715 million over the next decade.

Lawmakers and their top aides have been tinkering with various proposals — nudging here, trimming there — trying to find a deal that avoids added red ink for students and the government alike.

Democrats and Republicans met with Obama and Vice President Joe Biden on Tuesday at the White House. An outline of an agreement seemed to be taking shape Tuesday, with follow-up meetings Wednesday in Democratic Sen. Dick Durbin’s office yielding a final agreement.

Democratic Sen. Joe Manchin of West Virginia and Republican Sen. Richard Burr of North Carolina were the main negotiators, with Republican Sen. Lamar Alexander of Tennessee and Durbin filling the role of mediators. Sen. Angus King, I-Maine, Sen. Tom Coburn, R-Okla., and Sen. Tom Carper, D-Del., also were pushing the compromise to colleagues.

A senior administration official said the White House, too, was involved in the negotiations and that the White House supported the agreement. The official was not authorized to discuss the negotiations on the record and spoke on condition of anonymity.