The Premier League club are hoping to conclude a "loan with an obligation to buy" agreement in the coming days, according to Sky Sports Italia.
The 23-year-old will arrive at the Foxes for an initial loan fee of £2.7 million.
Leicester will then have to sign the player for £21.9 million when the season concludes, no matter what.
Under is left-footed and likes to play on the right wing.
The Turkey international has scored 17 goals in 88 games for Roma, but has not managed to secure a first team spot after showing much promise early in his career at the capital club.