The Daily Mail says the Gunners have offered Nketiah a five-year contract worth about ÂŁ100,000-per-week
The 21-year-old has scored 19 goals to help Nottingham Forest reach the Sky Bet Championship play-off final.
Nicolo Zaniolo's goal earned AS Roma a first major European title in more than 60 years after a 1-0 win over Feyenoord in the Europa Conference League final in Albania on Wednesday, completing a unique European trophy haul for Jose Mourinho.