Trump said last week that he planned to modify the asylum process to make it more hard for Central American migrants in the caravan to request protection.
The new restrictions are expected to be challenged in court.
Frequently attacking a caravan of thousands of Central Americans making their way north through Mexico, Mr Trump ordered troops to the border and declared the migrants to be an "invasion".
Administration officials have said existing USA asylum rules encouraged illegal immigration and bogged down legitimate claims.
Under the proposed rule changes, migrants who cross illegally would be ineligible for asylum, but they could still be spared from deportation by qualifying for a lesser status known as "withholding of removal".
United States asylum seekers would be given a hearing no matter how they arrived in the country until Mr Trump signed Friday's proclamation.
The current policy allows migrants - no matter how they entered the United States - up to a year to make an asylum claim.
Under the rule, migrants seeking asylum will have to make their claims at official ports of entry on the border.
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The president then moved to address the next questioner, NBC News White House correspondent Peter Alexander, who defended Acosta. Acosta chose to ask Trump why he seemingly "demonized" the immigrants traveling to the American border at the moment.
After failing to meet this goal, Trump said that the US was deploying 5,000 military personnel to the US-Mexico border.
Trump is expected to announce on Friday which countries the new rule will apply to, but critics say he is clearly targeting the small group of Central American migrants who are now traveling to the USA from Honduras, after escaping violence and unrest.
But directing the migrants to the official ports could create massive backups among applicants on the Mexican side of the border. Criminalization of asylum seekers rather than compassion has created some of the most shameful policies I've ever witnessed in this country.
U.S. Customs and Border Protection officials defend the practice on the grounds that the border crossings are not equipped to process hundreds of asylum seekers daily, and officers must continue to facilitate ordinary cross-border travel and trade while protecting the country from terrorists and drug runners.
Activists have accused border officials of unduly using this "metering" approach.
While there has been a recent spike in asylum claims, the number of illegal crossings at the southern border is a fraction of the totals seen from the late 1980s through the early 2000s. Those moves have been challenged in court. Before the midterm elections, Trump mulled ending birthright citizenship - the policy that ensures all children born on United States soil are automatically citizens - by executive order. The caravan is now around 600 miles from the United States.
About 4,800 migrants are sheltered in a sports complex in Mexico City, some 965 kilometres from the USA border. The migrants are largely poor people, and many say they're fleeing violence.
Similar caravans have gathered regularly over the years and have generally dwindled by the time they reach the USA border.