The President has it within his statutory and constitutional authority to suspend all grants of asylum, should he choose to do so. This was reaffirmed by the SCOTUS in June of this year, in the case, Trump v. Hawaii, No. 17-965 in the Supreme Court of the United States.
“8USC 1182(f) and 1185(a) By its terms, §1182(f) exudes deference to the President in every clause. It entrusts to the President the decisions whether and when to suspend entry, whose entry to suspend, for how long, and on what conditions. It thus vests the President with “ample power” to impose entry restrictions in addition to those elsewhere enumerated in the INA.”