"I'm not in favor of federalizing elections," Senate Majority Leader John Thune told reporters Tuesday, pointing to Constitutional requirements that states conduct their own elections.
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The president said in a Monday podcast interview that Republicans should "take over" elections in as many 15 states. Trump's calls come amid a push among Republicans in Congress to tighten voting requirements nationwide.
"I'm a big believer in decentralizing and distributing power," Thune added.
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However, the South Dakota Republican said he is supporting legislation known as the SAVE Act, which would require proof of citizenship when registering to vote in federal elections. The House has already passed the bill, but Republicans have not been able to overcome the 60-vote threshold required by the Senate's filibuster rules.