Those rates are currently 3.4 percent, but will double on July 1 without congressional action.

Under the GOP bill, interest rates would reset every year to a level equal to the 10-year Treasury note plus 2.5 percent. Under that formula, rates would stay below 6.8 percent for a few years, but are projected to rise above that rate several years from now.

The White House said Wednesday that the GOP bill is the "wrong approach."

"First, the bill would not guarantee low rates for today's students," the White House said in a Statement of Administration Policy. "Second, the bill's changes would impose the largest interest rate increases on low‑ and middle‑income students and families who struggle most to afford a college education."

Republicans based their bill on a proposal from the Obama administration's 2014 budget plan, and adds an 8.5 percent cap on the loan rate. But the White House said the proposal is different enough to warrant a veto.

"[T]he bill does not include the President's proposal to extend repayment options to borrowers who have already left school and often face the same debt burdens as current and future students," the statement said. "Finally, the administration believes that student loan interest rates should not be raised to reduce the deficit.

"If the president were presented with this legislation in its current form, his senior advisers would recommend that he veto the bill."