The US Food and Drug administration (FDA) could approve it for emergency use in October, the Financial Times reported, citing people familiar with the plan.
Mr Trump is lagging behind Democratic candidate Joe Biden in the polls, and getting a vaccine approved could be a way of boosting his numbers.
The US has the highest number of deaths related to coronavirus and the most cases out of any country, according to Johns Hopkins University.
Almost 177,000 people are confirmed to have died with the virus, while more than 5.7 million have been infected.
The White House declined to comment on the claims. AstraZeneca said that it had not discussed the possibility of fast-tracking approval with the US authorities, adding: “It would be premature to speculate on that possibility.”
The Oxford vaccine is considered one of the leading candidates among more than 100 teams vying to produce a vaccine. Researchers are currently conducting phase II and phase III trials, with results expected later this year.
Meanwhile, Mr Trump has also given emergency approval to the use of blood plasma taken from people who have already had Covid-19 to treat people currently sick with the virus.
Researchers are still investigating how effective the plasma is as a treatment. The theory is that antibodies from a survivor’s blood are already able to fight off the virus, so can help the current patient recover.
FDA chief scientist Denise Hinton said researchers would know more about plasma treatment in the months to come, but it should not be considered a new level of care for treating coronavirus patients.
Mr Trump said: “This is what I’ve been looking to do for a long time. Today I’m pleased to make a truly historic announcement in our battle against the China virus that will save countless lives.”