It's reported Monaco made an offer worth €35m and Arsenal went to €30m, but the Giallorossi want at least €50m for the Turkish talent.
Roma are under no pressure to sell Under, who joined them from Turkish side Basaksehir previous year.
The club sold Mohamed Salah to Liverpool last season for what now appears to be a bargain £34m.
Under has managed nine goals and two assists in all competitions in his debut season at the Stadio Olimpico.
Roma made it all the way to the semi-finals of the Champions League, and Under played his part in the tournament.
Left footed, the 20 year old has this year played on the right of a front three, on the right of midfield, and can also play as an attacking central midfielder or on the left.
The Gunners are pushing ahead with their transfer plans despite not yet having a new manager in place. Mikel Arteta is the strong favourite to take the job, according to David Hytner at the Guardian.
Roma will be keen to keep hold of Under this summer but if these fresh claims are to be believed, it will not be without fending off the interest of some other top European clubs.
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