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 FrelinghuysenHouse GOP picks two women to lead committees GOP struggles to find women to lead House committees Overnight Defense: NY/NJ bombings renew terror debate | US probes Syrian air strike | Senators push measure on Saudi arms sale MORE (R-N.J.), Frank WolfFrank WolfBottom Line 10 most expensive House races Benghazi Report and Hillary: What it means for Philadelphia MORE (R-Va.), Mike Simpson (R-Idaho), Kay GrangerKay GrangerTexas GOP's only female lawmaker calls on Trump to step down WHIP LIST: Republicans breaking with Trump GOP women break with Trump 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 BlackburnHouse votes to double budget for Planned Parenthood investigation Will Trump back women’s museum? The Hill's 12:30 Report 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.