Republicans across the United States have already embraced the president’s hard-line messaging about illegal immigration and crime. They are betting that they will benefit politically the way that Mr. Trump did during the 2016 presidential campaign.

At a rally on Saturday in Ohio, the president tried to bestow the issue on Troy Balderson, the embattled Republican House candidate in Tuesday’s special election. The contest — the final special election before the November vote — was notable for its surprisingly tough challenge from a Democratic opponent in an overwhelmingly Republican district.

“A vote for Troy’s opponent is a vote for open borders, which means massive crime,” Mr. Trump said at the rally. “We want our country to be a sanctuary for law-abiding Americans, not criminal aliens!”

The crowd responded with enthusiastic chants of “build the wall!”

On its own, the proposal for legal immigrants who accept government aid is unlikely to be a political driver in the midterm elections, said Jessica M. Vaughan, the director of policy studies for the Center for Immigration Studies, a group that supports stricter immigration policies.

“But I do think it will be well received by voters who tend to believe that our immigration system needs to be tightened, and that people applying for immigration benefits meet standards for self-sufficiency,” she said.

The proposed rule would allow immigration caseworkers to consider “heavily weighed negative factors” in determining whether green card candidates and other legal residents are likely to become dependent on government assistance, according to a leaked draft of the proposal. Those “negative factors” include accepting common tax credits and other welfare programs, like food stamps, transit assistance and health programs for children.

It would exclude some entitlements from being considered in the “public charge” determination — like immunizations, attending public school, receiving free or reduced-price school lunches or earned benefits such as disability insurance and Medicare.