The joint statement said presidents have the power to "suspend the entry of all aliens" and to impose "any restrictions he may deem to be appropriate on them" if they are judged to be "detrimental" to US interests under the Immigration and Nationality Act.
Accordingly, if the president issues a suspension or ban on entry through the US/Mexican border, those who illegally manage to enter the US will not be allowed to apply for asylum once there.
"Today, we are using the authority granted to us by Congress to bar aliens who violate a Presidential suspension of entry or other restriction from asylum eligibility," the statement said.
The rule will not apply retroactively. President Trump is expected to sign it shortly.