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Even though the majority of Congress members seem to support a fix, it still won’t be easy. Sen. Jeanne Shaheen, New Hampshire Democrat, introduced a bill to replace the cuts by closing offshore corporate tax loopholes — something Republicans took off the table before budget talks had even begun.

On the other side, Mr. Graham and Sen. Kelly Ayotte, New Hampshire Republican, suggested paying for the change by better regulating food stamp eligibility and child earned income tax credits to prevent fraud.