Bond producer Barbara Brocolli is said to be interested in hiring the director to helm the follow up to Quantum of Solace, The Sun reports.
A source said: "Danny is the man of the moment and he is being bombarded with offers from studios.
"He has been offered the chance to direct the 23rd Bond film by Barbara. The EON [Productions] team love his vision and think that it would work perfectly for the new look of Bond.
"Everyone on the EON team thinks Danny would be a fantastic addition to the Bond family."
The source added that the Bond franchise had moved "right up Danny's street" with the involvement of Daniel Craig and increased emphasis on characters and plot ahead of gadgets and one-liners.
Boyle shot to fame with his work on Shallow Grave and Trainspotting before directing The Beach and 28 Days Later and winning an Oscar for Slumdog Millionaire.
Craig and Bond co-star Judy Dench have both called for the reintroduction of Q and Moneypenny to the series when it returns.