In his memo, Mr Trump said the fee would not exceed the cost of processing the applications, but estimates have not yet been provided of what this may be.
Most Americans also feel that the USA political system is skewed to reward the powerful - 72 percent of US adults believe the American political system "works to benefit those in power" rather than "all people", the Post/ABC survey found.
The president also promised to fix asylum laws.
According to recent stats, more migrants seeking asylum have been denied their applications since Trump became president.
What did President Trump say?
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The announcement came as Trump blasted what he described as the United States' "weak, ineffective and dangerous" immigration laws on Twitter. Trump tweeted on Monday evening after the new guidelines were announced.
USA law already directs the Justice Department to finish asylum cases within six months, but with a backlog of more than 800,000 cases, asylum claims often take years to come to a conclusion.
The proposals Trump called for would significantly change the asylum system. "We've seen it over and over again, from turning away asylum-seekers at ports of entry, to trying to make those who enter between ports of entry ineligible for asylum, and most recently until now, sending asylum-seekers back to Mexico to wait for their court hearings", she says.
But with the proposed changes, Crow worries people may only be referred for asylum or withholding of removal proceedings - categories that have different standards and which aren't available to some people, including those who have previously been deported and then unlawfully reentered the country.
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The United States is facing a growing influx of migrants from the Central American countries of El Salvador, Guatemala, and Honduras with more than 1.4 million caught at the southern border after crossing illegally-half a million of them in the past 18 months.
As part of the memo, Mr. Trump is giving officials 90 days to come up with new regulations to ensure that applications are adjudicated within 180 days of filing, except under exceptional circumstances.
Whatever the final language of the proposed regulations, they are likely to be challenged in court.
And when people in those countries seek a better life, Crow says, "we're making it even more hard for them to reach safety once they flee".
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