The USA international is facing an uncertain future in North London on the back of his breakthrough season with Reims last term, where he netted 21 times in 37 Ligue 1 matches.
Balogun has been integrated into Mikel Arteta's squad during their pre-season friendlies, but he may find it difficult to secure regular starts in the Gunners' first team next season.
All of Gabriel Jesus, Eddie Nketiah and Leandro Trossard are battling for starting roles in the final third, while new signing Kai Havertz provides additional cover up top.
Balogun has already ruled out the prospect of another loan exit from Arsenal, who only have the 22-year-old under contract for another two years and could garner a healthy fee through a summer sale.
The likes of Marseille, Monaco, RB Leipzig and AC Milan are rumoured to be keeping tabs on Balogun's future, but Inter are reportedly at the head of the queue at this moment in time.
The Nerazzurri have already suffered a setback in their pursuit of Balogun, though, as Sportmediaset - via FC Inter News - claims that Arsenal have rejected their opening proposal.
The report states that Inter approached the Gunners with a fixed fee of €35m (£30m) with an additional €5m (£4.3m) in bonuses, but Arsenal are hoping to receive as much as £50m.
Inter's first bid is understood to be what the Nerazzurri were prepared to pay for Chelsea's Romelu Lukaku, who was widely expected to rejoin Simone Inzaghi's men permanently after his 2022-23 loan spell.
However, a return to San Siro for Lukaku now appears to be dead in the water, as Inter were allegedly left seething after learning that the Belgian was also in talks with Serie A rivals Juventus.
Marcus Thuram has joined on a free transfer from Borussia Monchengladbach to bolster the Nerazzurri attack, though, and Inzaghi is said to have a few alternatives to Balogun in mind.
West Ham United's Gianluca Scamacca and Udinese's Beto are among the strikers on Inter's shortlist, although a proposed move for Atletico Madrid's Alvaro Morata is not expected to come to fruition.
New York-born Balogun joined Arsenal's youth system in 2011, but the striker has only made 10 senior appearances for the club, scoring two goals and setting up one more.
Arsenal boss Arteta recently conceded that cuts would have to be made to his 30-strong playing squad - which the Spaniard labelled "unsustainable" - although only Granit Xhaka, Pablo Mari and Ainsley Maitland-Niles have been moved on thus far.
The Gunners round off their pre-season preparations in Wednesday's Emirates Cup tie versus Monaco before meeting Manchester City in the Community Shield on August 6.