Such is the excitement surrounding Under, the 20-year-old has already been touted by bookmakers as a potential Ballon d'Or winner.
"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 could go up against Barca when Roma face the Catalan giants in the Champions League quarter-finals next month, Monchi acknowledging the scale of the task facing Eusebio Di Francesco's side.
He said: "I wish I could know if we are going to eliminate Barcelona, but it is obviously difficult, we are facing possibly the best team in the world, or one of the best, but we have the [ambition] and we will try."