WASHINGTON, May 10 (Reuters) - Thousands of migrants are crossing into the United States this week before a new regulation takes effect that could bar most who cross illegally from seeking asylum, while others gathered on the Mexico side amid confusion about U.S. policy.

The U.S. rolled out a regulation on Wednesday that presumes most migrants are ineligible for asylum if they passed through other nations without seeking protection elsewhere first, or if they failed to use legal pathways for U.S. entry.

The new rule is a key part of President Joe Biden's border enforcement plan as COVID-19 restrictions - known as Title 42 - are set to end just before midnight on Thursday.

Under Title 42, which has been in place since March 2020, many border crossers were rapidly expelled to Mexico without a chance to seek asylum, leading to repeat attempts.

Homeland Security Secretary Alejandro Mayorkas said the new rule would mean harsher consequences for migrants crossing illegally who, if caught, could be deported and barred from the United States for five years if they do not qualify for asylum.

"We are making it very clear that our border is not open, that crossing irregularly is against the law and that those who are not eligible for relief will be quickly returned," Mayorkas said at a press conference in Washington.

Migrants have been amassing in Mexico near various parts of the border - many of them unsure about when, or how, to cross. Drone footage showed large crowds gathering at the border fence by El Paso, Texas, across from Ciudad Juarez, Mexico.

U.S. cities are expecting to receive some of those migrants after they cross the border. New York City said it is already receiving 500 per day and expects the number to increase after Title 42 expires on Thursday, leading Mayor Eric Adams on Wednesday to issue an executive order temporarily suspending a policy guaranteeing shelter for all those in need.