The 22-year-old scored 21 goals in Ligue 1 while on loan at Reims last season and Monaco are keen to bring him back.
The Gunners, however, want £50m for the American international, who remains a top target for Inter Milan after they pulled away from talks to sign Chelsea forward Romelu Lukaku.
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Inter are believed to be unable to meet that asking price but could insert a buyback clause or offer a favourable percentage of any future sale to help drive down the cost.
Balogun wants to play regular first-team football next season and that is unlikely to come at the Emirates Stadium, with Gabriel Jesus, Eddie Nketiah, Kai Havertz and Leandro Trossard all able to play up front.
Arsenal are also in talks over the sale of goalkeeper Matt Turner to Nottingham Forest.
The two clubs had been negotiating a fee of around £7m but the total package could now be worth in the region of £10m should it be completed.
Mikel Arteta is keen to bring in Brentford’s David Raya as a replacement and the two London clubs remain locked in talks over a fee.