Many are seeking United States asylum, but press reports say USA customs at some crossing points are only handling fewer than 50 applications a day.
US agents shot tear gas at the migrants after some tried to breach the fence separating the USA and Mexico.
REUTERS/Kim Kyung-Hoon Migrants, part of a caravan of thousands from Central America trying to reach the United States, rest as they wait at the El Chaparral port of entry border crossing between Mexico and the United States, in Tijuana, Mexico, November 23, 2018.
However, Olga Sanchez Cordero, senator from the Morena party and future Interior Minister (takes office on December 1) in the government of Mexican President Andres Manuel Lopez Obrador, has denied this media report. "His party controls the House and the Senate and it is on them", she said, arguing Trump should have worked more closely with Central American countries to prevent the caravans.
If Mexico were to assume "safe third country" status, asylum seekers would be required to claim refugee status in Mexico rather than the United States. US officials at the San Ysidro crossing generally accept no more than 100 asylum applications a day. He again threatened to close the country's southern border "if it becomes necessary".
No information was available on whether the woman was planning to claim asylum in the US.
The mayor of Tijuana blasted the government for leaving the city helpless as it struggles to cover the needs of some 5,000 caravan migrants.
She said Trump had chosen "to weaponize" and "politicize" the situation, but "has an opportunity here, especially with a new Congress coming in, to get this done".
"When they come back, we need to see how ... we can integrate them into an economic activity so that they can develop and not generate conflict with our own communities".
Trump took to Twitter again Saturday to reiterate that he plans to do away with the USA catch-and-release system, which allows asylum seekers to work and study sometimes for years while their cases are pending.
The man, who had previously been arrested in Florida and deported in June, told authorities he was part of the migrant caravan, according to the statement.
Last week a U.S. federal judge temporarily blocked the Trump administration from denying asylum to people who enter the country illegally. "Our very strong policy is Catch and Detain", he wrote in his Twitter message Saturday.