Mexican President-elect Andres Manuel Lopez Obrador suggested Sunday that the United States, Canada and Mexico work out a joint plan for funding development in the poor areas of Central America and southern Mexico.
"They're not coming into this country", said Trump, who has sought to make the caravan and border security in general a campaign issue for Republicans ahead of the US midterm elections.
"When I heard about the caravan, I knew it was my chance", he told the newspaper, noting his cousin and uncle still live in California and that he called them to let him know he was coming.
The migrants include many parents travelling with young children.
"The Mexican government is fully engaged in finding a solution that encourages safe, secure, and orderly migration", State Department spokesperson Heather Nauert said Saturday, "and both the United States and Mexico continue to work with Central American governments to address the economic, security, and governance drivers of illegal immigration". But people have been joining and leaving the caravan daily, some moving at their own pace and strung out in a series of columns.
Some boarded transport organised by the Guatemalan authorities at Tecun Uman, to return to Honduras voluntarily.
"The Mexican government should put the human rights of the caravan members at the heart of the response and not let the U.S. Trump administration's pressure prevent it from fulfilling its global obligations", said Erika Guevara-Rosas, the organization's director for the region.
Nevertheless, around 3,000 people were marching in the caravan on the Mexican side, according to an estimate from a federal police commander whose forces were closely monitoring the migrants' progress. "There is a lot of disorder", said Rivera, who was hoping to get on a bus back to Honduras laid on by the Guatemalan government.
They warned that anyone without papers would have to apply for refugee status or turn back, and anyone who crossed illegally would be detained and deported. At least a dozen of them jumped off the bridge into the Suchiate River.
President Trump last week warned Mexico that "we're calling up the military - not the Guard" if the government can not stop the caravan from reaching the border.
Mexican workers handed food bottled water to the migrants on the bridge.
The Mexican Interior Ministry said on Saturday that 640 Honduran migrants have requested refuge in Mexico.
"Let's all walk together!" and "Yes, we can!" they said.
Throughout Saturday, around 900 migrants - exhausted of waiting on the bridge - resorted to crossing the Suchiate River below on makeshift rafts and police did not intervene as they clambered up the muddy riverbank on the Mexican side.
A business administration graduate, Lopez said she couldn't find work in Honduras.
Some 2,000 Honduran migrants were already back home after giving up on continuing to Mexico, Guatemala's President Jimmy Morales said at a press conference in Guatemala City alongside his Honduran counterpart, President Juan Orlando Hernandez.
A migrant caravan whose numbers swelled overnight to an estimated 5,000 people at the Mexico-Guatemala border has resumed its march toward the US frontier.
"We are also are deeply concerned by the violence provoked by some members of the group, as well as the apparent political motivation of some organizers of the caravan". Children tried to make the best of the wet chaos.
The migrants pose a tough challenge to the Mexican government's pledge to stop the illegal travelers' plans to press ahead to the US border.
Earlier on October 20, the presidents of Honduras and Guatemala said around 2,500 migrants were repatriated to their respective countries.
In interviews along the journey, migrants have said they are fleeing widespread violence, poverty and corruption in Honduras.