In contrast, he said, the Jordan bill cuts "everything, indiscriminately, in a heavy-handed way."

Another Republican, Rep. Jo Bonner (R-Ala.), called the amendment is "misguided," and said cuts in the committee were done "surgically" with "deliberate intent."

"Across-the-board cuts are a lazy way to achieve something," added Rep. Dan Lungren (R-Calif.), and Rep. Bill Posey (R-Fla.) also spoke in opposition, saying the proposal would cut some national-security items. Reps. Rodney FrelinghuysenRodney Procter FrelinghuysenDems eyeing smaller magic number for House majority Puerto Rico mayor: Territory's profile has grown since hurricanes House panel advances homeland security bill with billion in border wall funding MORE (R-N.J.), Frank WolfFrank Rudolph WolfVulnerable Republican keeps focus as Democrats highlight Trump Bolton could be the first national security chief to prioritize religious freedom House votes to mandate sexual harassment training for members and staff MORE (R-Va.), Mike Simpson (R-Idaho), Kay GrangerNorvell (Kay) Kay GrangerBipartisanship alive and well, protecting critical infrastructure McCarthy's path to Speaker gets more complicated GOP senators visited Moscow on July 4, warned Russia against meddling in 2018 election: report MORE (R-Texas) and Jo Ann Emerson (R-Mo.) also spoke against the bill.

As expected, Democrats were also quick to pounce on the RSC proposal.

"This is a meat axe approach on top of a meat axe approach," Rep. Norm Dicks (D-Wash.) said, noting that the underlying bill already includes significant cuts. "It's a double meat axe approach."

Rep. Marsha BlackburnMarsha BlackburnElection Countdown: Takeaways from too-close-to-call Ohio special election | Trump endorsements cement power but come with risks | GOP leader's race now rated as 'toss-up' | Record numbers of women nominated | Latino candidates get prominent role in 2020 Top Koch official fires back at critics: We are not an 'appendage' of the GOP The Hill's Morning Report: Trump tries to rescue Ohio House seat as GOP midterm fears grow MORE (R-Tenn.) objected to Lungren's idea that the amendment was a "lazy" approach. "This government is over-spent, we have to get it under control," she said. Several freshman Republicans also spoke in favor of the measure.

Still, many House Republicans have already rejected more than $4 billion in proposed cuts to the spending bill throughout the week, which called into question whether enough Republicans would be able to support Jordan's amendment.

-- This story was updated at 1:30 p.m.