WASHINGTON The United States is already seeing high number of border encounters in certain areas, U.S. Homeland Security Secretary Alejandro Mayorkas said on Thursday ahead of a COVID-19 immigration restriction called Title 42's midnight expiration.

"Our borders are not open," Mayorkas told reporters at a White House briefing, warning of tougher consequences for those who try to cross the U.S. border illegally.

U.S. President Joe Biden's administration this week rolled out a new asylum regulation that could deny asylum to most migrants crossing the U.S.-Mexico border illegally. The measure will take effect immediately after Title 42 ends.

Migrant families caught crossing the U.S.-Mexico border illegally will generally be deported within 30 days if they fail an initial asylum screening under the new standard, Mayorkas said.

U.S. officials have previously said they aim to deport migrants in days if they are denied asylum under the new standard.

U.S. Customs and Border Protection (CBP) has in recent days been holding up to 28,000 migrants at its facilities, far beyond its stated capacity and in what appeared to be a record, Reuters reported on Thursday.

This week, the number of people registered crossing illegally surpassed 10,000 per day.

"I can't overstate the strain on our personnel and our facilities," Mayorkas said. "We could see very crowded Border Patrol facilities."

Republicans have criticized Biden for rolling back the hardline policies of Republican former President Donald Trump. A coalition of 22 Republican state attorneys general has opposed the new asylum rule for being "riddled with exceptions."

Some Democrats and immigration advocates say Biden's new regulation is too harsh and denies people their right to apply for asylum, comparing it to measures implemented under Trump blocked by U.S. courts.

The measure counters previous statements Biden made in 2020 on the campaign trail, when he said he thought it was "wrong" for people not to be able to seek asylum on American soil.

(Reporting by Jeff Mason and Steve Holland; Additional reporting by Ted Hesson; writing by Susan Heavey; editing by Costas Pitas and Aurora Ellis)