Graham is one of them — he told The Hill in early September that "If it's a demographic election, we're in trouble," and he and Sen. John McCainJohn McCainA Cabinet position for Petraeus; disciplinary actions for Broadwell after affair Meet Trump’s ‘mad dog’ for the Pentagon Wrestling mogul McMahon could slam her way into Trump administration MORE (R-Ariz.) called Romney during the primary and warned him to tone down his rhetoric on immigration.
On Thursday, Graham gave some hints at what a GOP version of comprehensive immigration reform might look like.
"Assimilation must be a major component in any immigration reform proposal," he said in one tweet.
"We must also address the issue of birthright citizenship our we will continue to have illegal immigration for as far as the eye can see. Without changes in birthright citizenship we will have future waves of illegal immigration," Graham continued.
It's unlikely that ending birthright citizenship will be acceptable to Democrats — but it's notable that Graham and other Republicans have ramped up pressure on their party to move on the issue.
"As a 14-year elected Republican official, I'm very concerned about the direction of the Republican party," Utah Attorney General Mark Shurtleff said on a Thursday conference call organized by National Immigration Forum, a pro-reform group. "It is an open opportunity right now to do something about this. My call to all elected officials is … to work on comprehensive immigration reform."
Shurtleff said he'd talked to GOP officials and expected there would be many conversations on the issue next week when Congress comes back in session.
"They need to step up and say 'we want to partner on this right now,' step up and propose your own [policies], work together on this and realize that that's what is going to be demanded," he said.
The group will host a two-day conference in early December to discuss how they can pressure Republican leaders to work with Democrats for reform.
One source close to Senate Republican leadership told The Hill on Wednesday that talks will start soon on the issue — but that actual policy proposals are unlikely until after the lame-duck session.
"My experience on elections is nothing really smart is decided on the day after the election," said the source. "I think moving too fast is going to hurt chances of getting something done. Something's going to happen sooner rather than later but if people start hashing out ideas this week or next that could lead to problems."
Graham authored a 2010 immigration reform bill with Democratic Sen. Charles SchumerCharles SchumerOvernight Finance: Trump takes victory lap at Carrier plant | House passes 'too big to fail' revamp | Trump econ team takes shape Anti-Defamation League: Ellison's past remarks about Israel 'disqualifying' Dems press Trump to keep Obama overtime rule MORE (D-N.Y.) . The proposal included strengthened border security, a system for admitting temporary workers and a path to legalization for those immigrants already in the country.