The 21-year-old was supposedly offered to the Foxes in a bid to soften the loss of Riyad Mahrez in a player-plus-cash swap deal earlier in the year, but the offer was initially refused by Leicester, who have now reignited their interest in the player once again.
According to the Mirror, Roberts will help Claude Puel fill the gap left by the potential departure of Mahrez, who is reportedly set to move to Manchester for a deal worth around £60m.
Roberts has had little success at Manchester City since moving from Fulham in 2015, but his subsequent two-and-a-half year loan spell with Celtic has allowed the player to show his qualities,
The youngster has had an impressive England youth career, scoring four goals in eight appearances for the England Under-20's side.
Should the young Englishman move to the King Power Stadium, he will join another promising youngster James Maddison, who signed from Norwich City for a fee of around £20m.
The Foxes have made a number of ambitious signings so far this summer, with Portuguese international defender Ricardo Pereira arriving from Porto, along with experienced central defender Jonny Evans, who joined in a cut price move from recently relegated West Brom.