The 19-year-old rose to prominence last season, scoring seven goals in 32 appearances as the team surprisingly finished second in the Turkish Super Lig, securing a berth in the third Champions League qualifying round.
Basaksehir had already said that City want to sign the forward, but it appears Pep Guardiola's side face competition from the Giallorossi.
"We're delighted that two giants of world football are interested [in] our player," Saral told the Anadolu Agency.
"Both Roma and Manchester City wants to sign Cengiz Under. They show their willingness for this transfer [to happen].
"Roma and Manchester City have made an official offer for him. We'll probably give our decision today about our player's future."