The Sun says Leicester had the young Hibees left-back watched in action during their Scottish Cup victory over Queen of the South on Monday night.
Doig, 18, has had scouts flocking to see him play throughout his debut season at Easter Road.
Now Brendan Rodgers' recruitment department at the King Power Stadium have started doing their own homework on the up-and-coming starlet.
Doig strolled through the cup clash with Queens at Palmerston in his latest impressive display.
Hibs have got Doig tied down on a long-term contract which runs until the summer of 2025.