(CNN) —  

Republican lawmakers have privately signaled to the White House that President Donald Trump should not nominate Ken Cuccinelli as the head of United States Citizenship and Immigration Services – or else he risks having another high-profile pick forced to withdraw, CNN has learned.

It was reported in late May that Trump was expected to tap the former Virginia attorney general for the Senate-confirmed role at the Department of Homeland Security after L. Francis Cissna resigned.

But in recent weeks members of the President’s party have been privately critical of the expected pick. People familiar with the discussions said they have made clear to the White House that Cuccinelli would face serious difficulty in being confirmed. Lawmakers have reminded officials in the West Wing that Cuccinelli, a former Virginia state attorney general, once led a political action committee that supported a primary challenge against Senate Majority Leader Mitch McConnell.

It’s unclear if the White House will take their advice. Officials have been sensitive to criticism that they do not vet their expected picks after Herman Cain and Stephen Moore both withdrew from consideration for spots on the Federal Reserve Board earlier this year. But one official said the White House’s decision will likely depend on whether lawmakers come out publicly against the nomination.

Several White House officials said they have envisioned Cuccinelli taking over for acting Department of Homeland Security Secretary Kevin McAleenan after his time in the post was up. McAleenan has signaled he does not want to be nominated for the permanent position, according to White House officials. He is standing in for former Homeland Security Secretary Kirstjen Nielsen, who left in April, amid a general shake-up at the department.

A Republican aide confidently predicted that Cuccinelli will not be confirmed.

There’s another major sign that Cuccinelli’s job prospect could be in trouble. USCIS Deputy Director Mark Koumans has assumed the role of acting director in the wake of Cissna’s forced-departure, which was led by Trump’s immigration adviser Stephen Miller.

In a letter to the workforce, Koumans said he anticipates that in the “coming days and weeks” there will be more information about the selection of the next director.

The head of a union representing USCIS employees also slammed Cuccinelli, warning his nomination could spell “the end of legal immigration as it currently exists.”

“It has become clear that the goal of this Administration is to end immigration all together. How better to do that then by appointing as the leader of USCIS someone who knows nothing about immigration, Adjustment of Status or Naturalization, and whose sole purpose is to destroy the agency that grants these benefits,” Danielle Spooner, the union president, said in a statement.

McAleenan declined to comment last week when asked about Cuccinelli’s potential role at USCIS and push back from the union.

A USCIS spokesperson said Tuesday that there was no update on Cissna’s successor at this time.