Per the source, Under will return to Roma after spending the season on loan at Leicester City.
The Foxes had the option to keep the Turkey international on a permanent move but will not sign Under.
Under has another two years remaining on his Roma contract but the Serie A side are prepared to part ways.
The winger struggled with injuries this season but has recovered and was called up to the Turkey squad ahead of Euro 2020 this summer.
Arsenal were linked with Under last summer but he ended up joining Leicester on a temporary move.
The 23-year-old has two goals and three assists in all 19 appearances in all competitions for the Foxes this season.
Under has eight goals and one assist for the Turkey national team, he has 26 international caps.