This agreement would change long-standing US policy and would place a strong barrier in the path of Central American migrants attempting to reach the United States "fleeing violence and harm", says David Leopold, past president of the American Immigration Layers Association, based in Washington, D.C.
Many have become desperate in recent days after taking stock of the asylum application process, The New York Times reported.
"Far from helping the caravan, they are hurting it", he told the Milenio television network.
"This is a vast majority of males seeking employment in the United States, not women and children", said Homan. "You've seen them, they're carrying signs, 'I want a job.' That doesn't qualify for asylum".
That could be because the incoming presidential administration, which does not assume office until Saturday, may not yet have the legal authority to conduct business with another country, says Stephanie Leutert, Director of the Mexico Security Initiative at the University of Texas at Austin.
For weeks before the 6 November congressional elections, Mr Trump raised the alarm about the migrant caravans and ordered some 5,800 United States troops to the border to support border patrol agents. A young woman fell to the ground unconscious, and two babies cried, tears streaming from the gas.
Or that Mexico's Interior Ministry said around 500 migrants tried to "violently" enter the US, the Associated Press reported, adding that it would immediately deport those people and would reinforce security.
A group of migrants climbed a fence as they tried to reach the US-Mexico border
"There were some press members who helped throwing water to the children and there was a child who was hardly breathing and a person from the press grabbed him and took him away", she said in Spanish.
"Mexico should move the flag waving Migrants, many of whom are stone cold criminals, back to their countries", Trump tweeted.
Mexican officials said the migrants had taken part in "violent" chaos, which originally began as a peaceful march.
Irineo Mujica, who has accompanied the migrants for weeks as part of the aid group Pueblo Sin Fronteras, said the aim of Sunday's march toward the US border was to make the migrants' plight more visible to the governments of Mexico and the U.S. "Congress, fund the WALL!".
Mexican police, some wielding shields, scattered migrants and protesters, many of whom rushed toward the US border.
On Monday, pedestrian crossings were suspended at the San Ysidro port of entry at both the east and west facilities, and all northbound and southbound traffic was halted.
"They want us to wait in Mexico but I for one am desperate. I think there is no chance", he said.
He said that it was a "very risky situation" for agents and they responded appropriately.
Trump wants migrants to remain in Mexico while their USA asylum cases are adjudicated.