Presidential pardons are usually issued to people who have been convicted.
Pre-emptive pardons, for people who have yet to be convicted or charged with any crime, are rare but have been issued.
In the most famous example President Gerald Ford pardoned Richard Nixon after he resigned following the Watergate scandal. At the time Nixon had not been charged.
President Jimmy Carter also issued an unconditional pardon to hundreds of thousands of men who evaded the Vietnam draft.
The report came as pardons became central to Trump's final weeks in office.
Last week, he pardoned Michael Flynn, his former national security adviser, who pleaded guilty to lying to the FBI over his discussions with a Russian ambassador.
In response to the suggestion of a pre-emptive pardon, Robert Costello, a lawyer for Giuliani, told the New York Times that his client "didn't do anything wrong".
A spokesman for Giuliani added: "Mayor Giuliani cannot comment on any discussions that he has with his client [the President]."
Trump is expected to issue numerous pardons before he leaves office, including some of his former aides who were convicted following the investigation into links between his 2016 campaign and Russia, which Trump has repeatedly called a Democrat "witch hunt".
Academics are increasingly arguing over the issue of whether Trump could issue a pre-emptive pardon to himself.
Allies of the President have encouraged him to do so, suggesting that he will face politically motivated legal attacks after he leaves the White House.
There is no precedent for a president pardoning themselves, and polls have suggested that a large majority of Americans believe they should not be able to do so.
