The New York Post qualified that report, indicating that Mexico might be backing out of the agreement.
The deal was seen as a way to dissuade thousands of Central American migrants from seeking asylum in the USA, a process that can take years.
Only a small minority of Central American applicants are ultimately granted political asylum in the United States, but the decision making progress can take months or years - during which time many are released and gain footholds in the United States.
That changed this week as Donald Trump claimed all migrants will stay in Mexico until their claims are processed.
Trump did turn to Twitter on Saturday evening to discuss the migrant matter, vowing that those at the U.S. -Mexico border wouldn't be permitted into America "until their claims are individually approved in court".
"The medium- and long-term solution is that people don't migrate", she told the newspaper.
Reuters noted that Sanchez "did not explicitly rule out" that Mexico could allow Central American caravan migrants ― thousands of whom have arrived in Tijuana, just south of California, in recent days ― to wait in the country while their claims are processed in the U.S. Sanchez did, however, say that plans for Mexico to assume "safe third country" status had been "ruled out". Mexico can not be considered safe for asylum seekers, many argue.
The White House did not comment on the reported deal on Saturday.
"There is no agreement of any sort between the future Mexican federal government and the USA", the incoming interior minister said in a statement. "All will stay in Mexico".
Ms Sanchez Cordero ruled out that Mexico would be declared a "safe third country" for asylum claimants.
After a trek of more than a month from Honduras, almost 5,000 migrants - including women and children - are now in Tijuana living in a makeshift shelter.
"They're going to have to open the borders [for the migrants] to put in the request", Vences said.
It added that under the new rules, an applicant whose asylum claim is denied would not be allowed to return to Mexico but would remain in U.S. custody pending immediate deportation to his or her home country.
Mr Trump also said the United States would consider closing the southern border if the administration deemed it necessary, and has said he would use lethal force on the migrants. "Our very strong policy is Catch and Detain", he wrote in his Twitter message Saturday. USA officials at the San Ysidro crossing generally accept no more than 100 asylum applications a day.
But according to The Associated Press, the new Mexican leadership was telling a somewhat different story.
Earlier this month, a federal judge in California halted the Trump administration's latest attempt to seal the USA southern border by barring migrants from seeking asylum inside the country.