WASHINGTON — In a bipartisan rebuke, the Senate voted on Wednesday to overturn a major Trump administration rule that would sharply limit debt relief for students misled by schools that lured them in with false claims about their graduates’ career and earning prospects.

In a 53-to-42 vote that included 10 Republicans, the Senate struck down a revised Education Department rule completed in September by the department’s secretary, Betsy DeVos. The House passed a companion resolution in January. The legislation will now go to President Trump, who will decide whether to uphold the rule with a veto or side with Congress over his own education secretary.

He has told Senate Republicans he is “neutral” on repealing the rule, though he has yet to comment on his veto intentions.

If Mr. Trump does not sign it, Democrats say they will press for a veto override. Senator Richard J. Durbin, Democrat of Illinois and the minority whip, successfully recruited Republicans for Wednesday’s vote after veterans groups — which he called a “critical ally” — backed his resolution.