Unless Congress acts by July 1, the rate will rise to 6.8 percent.
Courtney believes a deal can be ironed out, though he said Congress might have to pass it next month and make it retroactive.
Both the House and Senate will adjourn for July 4 recess at the end of this week.
A bipartisan deal has been ironed out in the Senate, but that bill — backed by Sens. Richard Burr (R-N.C.), Joe Manchin (D-W.Va.) and others — does not have the support of Senate Majority Leader Harry Reid (D-Nev.).
The Senate this year has not passed a student loan measure, but the House has cleared legislation that embraces a market-based approach to rates. Obama threatened to veto the House bill, which passed mostly along party lines.