"I'm not going to make any bold claims today," Republican Study Committee Chairman Rep. Mark Walker (R-N.C.) told The Hill, "but there will be further discussions ... members will not look the other way on this vote."

Frelinghuysen, who chairs the powerful Appropriations Committee, was among 12 House Republicans to vote against the GOP's tax measure earlier. He also voted against the original House version of the tax plan last month.

Frelinghuysen and most of the other GOP defectors on Tuesday represent districts in high-tax states that are expected to be negatively impacted by the bill’s limits on the state and local tax deduction.

Walker said that as a committee chairman "there is an agreement that you support the will of the conference — with him trying to vote against the will of the conference, that's going to be a problem."

Walker wouldn't say how the issue would be resolved, only that "further discussions are needed."

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When asked who would make a good replacement for Frelinghuysen as Appropriations chairman, Walker suggested conservative Rep. Tom Graves (R-Ga.), calling him "one member who does a great job on Appropriations."

Watch the video above to hear Walker in his own words.