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The Gunners have been strongly linked with a move for the 21-year-old and have reportedly been watching over the forward in recent weeks.
Italian publication Tuttomercatoweb now claim Arsenal submitted an offer of around £29million for Under, who has scored and created five goals in 19 matches this season for the Serie A club.
However, the offer from is reported to have been instantly turned down by Roma, who value the Turkey international closer to the £45million mark.
It’s understood Roma do not wish to sell Under in January and are in no financial predicaments to do so following progression into the Champions League last 16.
It remains to be seen if Arsenal will make a second bid for the in-demand forward, but Roma will be more likely to do business at the end of the season rather than next month.