WASHINGTON—Republican lawmakers showed some openness Sunday toward the idea of eliminating tax breaks, a crucial step if lawmakers are to reach an agreement on cutting the deficit and break the current impasse over raising the federal debt ceiling.

Republican Sen. John Cornyn of Texas said lawmakers should close some tax loopholes and eliminate breaks as part of a broader effort to cut rates, though he cautioned that there may not be enough time to include such changes in a budget deal now under negotiation.

"Do...