SAN DIEGO — Asylum-seekers whose claims were dismissed or denied under a Trump administration policy that forced them to wait in Mexico for court hearings will be allowed to return for another chance at humanitarian protection, the U.S. Homeland Security Department said Tuesday in a move that could benefit tens of thousands of hopefuls.

Registration will begin Wednesday for asylum-seekers who were subject to the “Remain in Mexico” policy and either had their cases dismissed or denied for failing to appear in court, The Associated Press has learned.

Under that criteria, it is unclear how many people will be eligible to be released into the United States pending a decision on their cases, according to a senior Homeland Security official who spoke on condition of anonymity because the announcement had not been made public.

But Michele Klein Solomon, the International Organization for Migration’s director for North America, Central America and the Caribbean, told the AP that she expected at least 10,000 people. Her organization is working closely with the Biden administration to bring people to the border and ensure they test negative for COVID-19 before being allowed into the country.

The estimate seems low. Syracuse University’s Transactional Records Access Clearinghouse said the move could benefit 34,528 migrants — 27,842 who denied for failing to appear in court and 6,686 whose cases were dismissed.

A group of migrants mainly from Honduras and Nicaragua wait along a road after turning themselves in at the U.S.-Mexico border in La Joya, Texas.

Many are believed to have left the Mexican border region, thinking their cases were finished, raising the possibility that they will make the dangerous trek to return. The official said the administration is aware of those dangers and considering bringing people to the United States, like it is doing to reunite families that remain separated years after Trump’s “zero tolerance” policy on illegal border crossings.

The move is another significant effort at redress for Trump policies that Biden administration officials and their allies say were cruel and inhumane and defenders say were extremely effective at discouraging asylum-seekers from coming to the U.S.