Cited were concerns raised by a number of Republicans, including former presidential candidate Mitt Romney, that threatening a government shutdown could cause Republicans to lose control of the House in the 2014 elections.
But Lee spokesman Brian Phillips said the 14 senators who have signed on to the effort are just a start.
He said FreedomWorks and four other tea party and conservative groups are urging activists around the country to show up at their congressional town hall meetings to demand backing for the defunding proposal.
Also, Phillips said, there will be a "big push" in Washington, D.C., once Congress returns in September, including rallies. Lee has said in fundraising emails that despite critics, "the American people are overwhelmingly on our side."
A recent poll of Utah political insiders by Utahpolicy.com and KSL found that 46 percent of Republicans believed Lee's effort made him and his supporters "look foolish and another 6 percent said it "damaged the party by dividing lawmakers."
University of Utah political science professor Matthew Burbank said he expects Lee will find support in the town hall meetings even though that may not reflect the majority of voters.
"What we've seen in probably the majority of these town hall meetings is by and large the people who are motivated to attend are more supportive," Burbank said. "The position he has staked out on that is not clearly supported in his own party."
None of the meetings Lee is holding are in Salt Lake County, the state's largest and most liberal county.
"You'd be much more likely to run into opposition if you did one in Salt Lake County," Burbank said.
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