Incoming Interior Minister Olga Sanchez Cordero also told news agency Reuters no agreement had been reached - although the Washington Post article initially quoted her as saying it was a "short term solution" to deal with the migrant caravan.
The Post had quoted Sanchez herself as saying the administration of Mexico's President-elect Andrés Manuel López Obrador, who will take office on December 1, had "for now" agreed to the so-called "Remain in Mexico" plan.
The possible deal shows that the Trump administration has, is about to, overcome Mexico's historic reticence to deepen cooperation with the USA on an issue widely seen there as America's problem.
President Trump tweeted late Saturday that no one will be allowed to cross the border into the USA without being properly processed, and vowed that he would close the entire southern border with Mexico if the situation there gets out of hand.
According to the Washington Post, the two sides have agreed in principle that asylum seekers will wait in Mexico for courts to assess their claims-and anyone denied gets sent back to their home country. If for any reason it becomes necessary, we will CLOSE our Southern Border. He also admonished the US central bank over the cost of borrowing money. "All will stay in Mexico", the president wrote.
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, and activists have long argued Mexico does not have the security conditions to offer safe haven for Central American migrants fleeing poverty and violence.
The White House did not immediately return a message seeking comment.
The administration of Mexico's current president, Enrique Pena Nieto, rejected a similar Trump administration proposal past year.
After months on the road, the men, women and children in the camp are now living on donated food, sleeping on blankets and sharing makeshift toilet facilities.
American officials will be able to process at least twice as many asylum claims under the new system because they would not be limited by detention space at U.S. ports of entry, the Post report said.
"We only will allow those who come into our Country legally".
Stephanie Leutert, director of the Mexico Security Initiative at the University of Texas at Austin, told AP the Remain in Mexico plan is part of a strategy to take away the ability of migrants to live and work in the USA while cases are processed.
Meanwhile, the government of the state of Baja California has identified 7,000 jobs for which migrants could possibly earn income while they await hearings in the U.S. But agents at the San Diego port of entry process fewer than 100 claims per day. Other than that our very strong policy is Catch and Detain.
He has expressed determination to stop the caravan from reaching the USA, deploying troops to the border and warning of consequences should countries assist the caravan in its journey.
Tijuana Mayor Juan Manuel Gastelum on Friday declared a humanitarian crisis in his border city of 1.6 million, which he says is struggling to accommodate the influx.