WASHINGTON — It’s an image many Americans would find rather upsetting: a recently laid-off millionaire, luxuriating next to the pool eating grapes bought with food stamps while waiting for an unemployment check to roll in.

Under the Republican bill to extend a payroll tax holiday scheduled to be voted on in the House as early as Tuesday, those Americans with gross adjusted income over $1 million would no longer be eligible for food stamps or jobless pay, producing $20 million in savings to help pay for the tax cut for American workers. The idea is also embraced by many Democrats, who had a similar version of the savings in a Senate bill to extend the payroll tax cut, as did a failed Republican Senate bill.

Yet as it turns out, millionaires on food stamps are about as rare as petunias in January, even if you count a lottery winner in Michigan who managed to collect the benefit until chagrined officials in the state put an end to it.

But the idea of ending unemployment insurance for very high earners — which would be achieved essentially through taxing benefits up to 100 percent with a phase-in beginning for those with gross adjusted income over $750,000 — demonstrates an increasing desire among members of Congress to find some way to make sure that the wealthiest Americans contribute more to reducing the deficit and paying for middle-class tax relief.