Weeks after Congress missed a July 1 deadline to prevent interest rates on federal student loans from doubling, a bipartisan group of senators said this week that they finally had a deal.

Under the compromise, undergraduates would pay 3.86 percent interest on both subsidized and unsubsidized federal loans in the upcoming academic year. After that, future interest rates on Stafford loans would match the yield of a 10-year Treasury note, plus an additional 2.05 percentage points. Those rates would be capped at 8.25 percent.

For graduate students' loans, the deal would add 3.6 points to the Treasury rate, with a 9.5 percent rate cap.

"While this is not the agreement any of us would have written and many of us would like to have seen something quite different, I believe that we have come a very long way on reaching common ground," Sen. Dick Durbin (D-Ill.) said.

Congress had a full year to hammer out a deal after passing a stopgap fix last summer that extended a flat 3.4 percent interest rate until July. Yet House Republicans, Senate Democrats, and the White House all favored competing longer-term solutions this time around. When neither party budged by the deadline, the interest rates spiked back up to 6.8 percent.

That change was projected to cost the average borrower an additional $2,600 over the life of a 10-year loan.

Some Democrats had strongly opposed tying student loans to Treasury notes, which have a varying interest rate. Should the interest on those notes spike, students could get stuck with much higher bills once they leave school.

Yet after a meeting Tuesday with President Obama, in which the president urged senators to end the impasse, Democrats ultimately relented.

The House already passed a similar bill back in May that would tie interest rates to the 10-year note, plus 2.5 percent points, with a cap at 8.5 percent. Speaker of the House John Boehner (R-Ohio) said he had yet to see the full Senate deal, but that the basic similarities between the two measures were promising.

"When we see the details, I am hopeful that we will be able to put this issue behind us," he said.

In the Senate, Majority Leader Harry Reid (D-Nev.) said a vote could come as early as this week.