A bipartisan group of senators is plodding ahead with talks on comprehensive immigration overhaul legislation, but Republicans say they’re reluctant to sign on to any agreement unless the White House takes stronger action to stem migration at the U.S.-Mexico border.

Republicans also want to see the Biden administration restart construction on the border wall before they’ll consider legal pathways that Democrats want for undocumented immigrants living in the United States.

“I don’t think they’re going to have a snowball’s chance in hell of doing that given the massive influx at the border,” said Sen. Lindsey Graham, R-S.C. “There’ll be no immigration reform until you get control of the border.”

The group of lawmakers — which usually includes Graham and Sens. Richard J. Durbin, D-Ill., and John Cornyn, R-Texas, among others — was next scheduled to meet late Wednesday afternoon, said Sen. Bob Menendez, D-N.J., one of the main negotiators.

Menendez is the main sponsor of White House-backed legislation that supports a pathway to citizenship for more than 11 million undocumented immigrants.

“So far, for our Republican colleagues, the focus has been all about the border,” Menendez said. “And we certainly are willing to deal with questions about the border — but the question is much broader. What do we get for that? That’s an open question.”

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The White House has been under fire from Republicans for weeks in the wake of historically high migration numbers, which conservative lawmakers blame on Biden’s undoing of stringent Trump-era policies, including the use of Title 42, a public health directive, to turn away unaccompanied children at the border.

Democrats, meanwhile, say the spike in migration has more to do with endemic “push factors” such as food insecurity, violence and climate change in Central American countries.

The Biden administration has made significant progress in reducing the number of migrant children being held in U.S. Customs and Border Protection custody, although thousands remain in Department of Health and Human Services care.

But new numbers released by CBP late Tuesday suggest that the high migration rate is continuing: More than 111,000 single adults were encountered and expelled by border agents last month, a roughly 11 percent increase from March. Border agents also encountered more than 17,000 children who arrived without their parents in April, a slightly lower number than in March.

Democrats have weighed other avenues for making headway on the immigration overhaul their party’s base desperately wants. Some lawmakers have floated the idea of including immigration provisions in a future bill passed through budget reconciliation, with no Republican support, although it’s unclear whether the Senate parliamentarian would greenlight such a move.

[Democrats weigh reconciliation bill for immigration action]

“We need to look at every legislative path possible to get comprehensive immigration reform done — including through reconciliation,” Sen. Patty Murray, the No. 3 Democrat in the Senate, said in a statement last week.

The House passed two bills in March to provide a pathway to citizenship for undocumented immigrants brought to the U.S. as children, temporary protected status holders and undocumented agricultural farmworkers. But those measures are unlikely to earn 60 votes in the narrowly divided Senate.

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Some Democrats signaled optimism about the bipartisan meetings. Although comprehensive immigration overhaul has remained elusive for years, lawmakers came close in 2013 when the Senate passed a bipartisan bill that was not taken up in the House.

“They’re going well,” Sen. Catherine Cortez Masto, D-Nev., said of the negotiations. “They’ve been productive so far.”

Bipartisanship hasn’t been totally out of reach during the 117th Congress: Cornyn and Sen. Kyrsten Sinema, D-Ariz., introduced a bill in April to address the recent influx of migrants at the U.S.-Mexico border by ramping up staffing at immigration agencies and streamlining immigration court proceedings.

But Cornyn, who has been a champion of efforts to provide citizenship to undocumented immigrants brought to the U.S. by their parents, hinted that movement on broader legislation was proving difficult.

“A lot of talk,” Cornyn said. “No progress.”