The Foxes are said to be hopeful of finalizing a deal with Roma that would see Under join them on an initial loan with an obligation to buy for €27 million (£24.6 million) after the season.
Leicester City apparently offered Under a five-year deal. His move to King Power Stadium should be made official in the coming week barring any unforeseen circumstances, Romano notes.
A move for the Turkish winger comes after the Foxes failed to sign Francisco Trincao from Barcelona earlier this summer with a bid reportedly worth €40 million.
The 23-year-old Under was once considered a big part of Roma's future. However, he was reduced to a bench role at the end of last season after being plagued by injuries at the start of the campaign.