"He said I'm not supporting it," Huelskamp, one of the House's more conservative members, said of Cantor, after Republicans convened a special New Year's Day caucus conference to consider the way forward for the legislation.
"I took it as, it's a violation of our conservative principles."
Several House Republicans emerged from the meeting saying they would not support the bill as it was, with congressman Spencer Bachus saying he would be "shocked" if it did not return to the Senate with a demand that more spending cuts be inserted.
Lawmakers are under pressure to pass at least some form of fiscal cliff crisis legislation before financial markets open on Wednesday, but it was increasingly unclear whether that could be done.
Huelskamp said House Speaker John Boehner laid out the legislation in a discussion with members, but did not express his overt support or opposition to the bill.
In a statement, Boehner's office acknowledged there were deep concerns within his party about the legislation.
"The speaker and leader laid out options to the members and listened to feedback. The lack of spending cuts in the Senate bill was a universal concern amongst members in today's meeting," Boehner spokesman Brendan Buck said.
"Conversations with members will continue throughout the afternoon on the path forward."