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House Majority Leader Eric Cantor, Virginia Republican, said Wednesday’s Senate votes rejecting both proposals on the table will “take away two numbers” from the discussion, though he also said it’s premature for House Republicans to abandon their position of higher cuts.

Earlier this year, rank-and-file House Republicans pressed their leadership to dig deeper on cuts, and on Tuesday Rep. Tom Graves, a Georgia Republican who helped organize that push, said his colleagues remain firm.

“I know the House is going to stand strong. What’s awesome about this freshman class is that they’re focused on the next generation and not the next election,” Mr. Graves told the Washington Times-affiliated “American’s Morning News” radio program. “They’re willing to put up the tough votes, because that’s what it’s going to take.”

David Eldridge contributed to this article.