(CNN) US President Donald Trump and Mexico's top immigration officials painted sharply different pictures Tuesday of the large groups of Central American migrants at the border.

On Twitter and during a live televised meeting in the Oval Office, Trump said migrants in the so-called caravan he'd been warning about for weeks were either returning to their home countries or staying in Mexico.

Despite the large Caravans that WERE forming and heading to our Country, people have not been able to get through our newly built Walls, makeshift Walls & Fences, or Border Patrol Officers & Military. They are now staying in Mexico or going back to their original countries....... — Donald J. Trump (@realDonaldTrump) December 11, 2018

"A lot of the people that wanted to come into the country. And really they were going to come in no matter how they wanted to come in. They were going to come in, even a rough way. Many of these people are leaving now and going back to their countries. ... If you notice, it's getting a lot less crowded in Mexico," Trump said. "And a lot of them are going to stay in Mexico, and the Mexican government has been working with us very well. So we appreciate that. But they haven't been coming into our country. We can't let people come in that way. So that's pretty much it."

But Mexican immigration authorities said Tuesday that many migrants were still waiting at the border for the chance to make their case for asylum in the United States -- and that more than 1,000 migrants from the caravan may have crossed into the United States already.

According to data provided by the US Department of Homeland Security, US Customs and Border Protection had apprehended 1,036 caravan migrants as of Monday. And more than 5,000 migrants were waiting to seek asylum in a queue at the San Ysidro port of entry.