Senate Republicans are discussing a legislative strategy to break the U.S. budget stalemate that would let Congress extend tax cuts for all except the highest income levels, said two Republican aides who spoke on condition of anonymity.

The Republican-controlled House would vote on two separate bills, the aides said. One would extend tax cuts at all income levels. That would have wide support among Republicans though President Barack Obama has said he won’t accept it.

The other bill would allow tax cuts for top earners to expire. Democrats would support that plan and Republicans would be likely to provide enough votes to pass it in the House, one aide said. The Democratic-controlled Senate would pass and send that measure to Obama, the aide said.

Such an approach would let Republicans go on record in support of their proposal even as enough of them would join Democrats to pass Obama’s plan.

Obama and House Speaker John Boehner haven’t reached an agreement that would incorporate tax-cut extensions into a broader budget deal. If Congress doesn’t act, more than $600 billion in tax increases and spending cuts will take effect starting in January.

Boehner hasn’t ruled out extending tax cuts for all except the top earners, though he has insisted that all tax reductions should be extended.

Don Stewart, a spokesman for Senate Minority Leader Mitch McConnell of Kentucky, said in an e-mail today that “Republicans will continue to look for ways to protect American families and jobs while strengthening entitlement programs and continuing to advocate for the types of intelligent reforms in Washington spending that the president has yet to propose.”

McConnell “does not advocate raising taxes on anybody or anything,” Stewart said.