Meanwhile, Trump officials said they were eager to receive the vaccine and would take it when offered.
However, some said they were concerned it would deliver a wrong message, making it appear as if Trump staff members were cutting lines to protect the president.
Trump has contracted the virus during his campaign for the presidential elections. He has now recovered from the virus.
Pfizer-BioNTech's first doses of vaccine left a facility in Michigan early Sunday, with United Parcel Service and FedEx working together to ship the vaccine to all 50 states for distribution.
However, a White House spokesman did not respond to request to comment on the issue.