Asylum seekers hoping to enter the U.S. via its southern border will have to wait in Mexico while they are assessed, President Donald Trump announced Saturday, appearing to confirm a report about a bilateral deal published by The Washington Post.

The move was cautiously welcomed by some refugees currently at the border, even as Mexico's incoming interior minister Olga Sanchez Cordero, who was quoted by the Post as confirming the agreement, later issued a denial.

"Migrants at the Southern Border will not be allowed into the United States until their claims are individually approved in court," Trump wrote on Twitter.

He added that the U.S. "will allow those who come into our Country legally" and emphasized: "All will stay in Mexico."

The deal, which would overhaul U.S. border policy, comes with Trump outraged over the presence of thousands of Central American migrants who marched to Mexico's border city of Tijuana hoping to enter the U.S. for a better life free from poverty and gang violence in their homelands.

"For now, we have agreed to this policy of Remain in Mexico," the Post quoted Sanchez Cordero as saying. The government of new President Andres Manuel Lopez Obrador will enter office on December 1.

Caravans of migrants, many of whom say they are fleeing from persecution, poverty and violence in Honduras, Guatemala, and El Salvador, were making their way through Central America towards the U.S. border for weeks. Many of them want to claim asylum.

Several thousands of migrants, including women and children, are reportedly in the Mexican border city of Tijuana living in a makeshift shelter.

Trump has sent almost 6,000 soldiers to the Mexican border in support of Customs and Border Protection agents and National Guard troops already there, to forestall what Trump has called an "invasion" by "very bad people."