Kenneth T. Cuccinelli II, the immigration hard-liner who was expected to be President Trump’s pick to coordinate immigration policy, will instead be chosen to take over for the embattled official who has overseen the legal immigration system, according to two people briefed on the situation.

The official, L. Francis Cissna, whose role as the head of United States Citizenship and Immigration Services has included overseeing a visa system that many White House aides view as broken, has submitted to pressure to step down, the two people said.

Mr. Cuccinelli, a former Virginia attorney general, is expected to be tapped to replace Mr. Cissna, the two people said. Mr. Trump had asked Mr. Cuccinelli in recent days to help coordinate policy across agencies, akin to the “immigration czar” job that the president has considered creating for months. The move startled officials at the White House and at the Department of Homeland Security, where one West Wing official said Mr. Cuccinelli would work.

For now, he will be used to move out Mr. Cissna, the people familiar with the move said.

But Mr. Cissna, who was supported by a number of immigration restrictionists, held a Senate-confirmed role. And people close to the Senate majority leader, Mitch McConnell, who has been a target of Mr. Cuccinelli’s in the past, said that the former attorney general’s chances of being confirmed were close to zero, creating immediate questions about the next steps in the process.