Under made his professional debut in 2014 at Altinordu, playing 51 times in the Turkish second tier before earning a move to Super Lig side Bursaspor. Nine goals in 43 games in his debut season there saw Roma splash out €13.4 million (£11.8m/$15.5m) for the right winger.
He could now be on the move again, with Bayern Munich looking to find long-term replacements for Arjen Robben and Franck Ribery, who are due to turn 35 and 36 this season respectively.
Under is a regular at Roma and after impressing there in Serie A and the Champions League, is in demand across Europe. He has been linked with clubs such as Monaco and Arsenal, but Altinordu's chairman says that Bayern could be a likely destination for the €60m-rated (£53m/$70m) 21-year-old.
“Bayern Munich are keeping a close eye on Cengiz,” Mehmet Ozkan told Gazzetta dello Sport.
“All I can say is that he could move for a lot of money.”
Bayern could face a lot of competition for the 15-time Turkey international, with Roma sporting director Monchi confirming Barcelona interest in March. However, Monchi insisted that he was not for sale and that the Italian club had long-term plans for the winger.
"It's true that Barcelona have an interest in the player," Monchi said in an interview with Radyospor.
"Bayram Tutumlu [Under's agent] has a strong relationship with Barca and he informed me that there is definitely an interest there.
"We have not received any offers for the player but it would not surprise me if he was being tracked – he is a young, talented player who is scoring a lot of goals.
"Our aim is for Cengiz to become a very important player for us because he has youth and all the qualities needed to do that."
Under has started five of Roma's Serie A games so far this season, scoring one goal. He also netted in their 5-0 Champions League win over Viktoria Plzen and started the 3-0 loss to Real Madrid in their opening game.