Osimhen's name emerged during separate talks between Arsenal sporting director Andrea Berta and Galatasaray over the Turkish champions' interest in Gabriel Martinelli and Ethan Nwaneri, for whom Galatasaray have tabled an offer of around £38.4m, according to The Sun.
It remains unclear whether Arsenal raised the possibility of signing Osimhen or whether the 27-year-old was offered as part of the wider discussions.
What is clear is Galatasaray's resistance: as recently as this week they turned down a verbal offer worth £111m (€130m) from Saudi Arabian side Al-Hilal, making clear they have no intention of negotiating a sale at any price this summer.
Osimhen is not currently Arsenal's leading striker target. That remains Atletico Madrid's Julián Álvarez, though his £100m-plus valuation and reported preference for a move to Barcelona have complicated that pursuit and pushed the Gunners toward alternatives.
Arsenal have a long-standing interest in Osimhen dating back to when Edu was the club's sporting director, and the striker also came close to joining Chelsea before his move to Turkey. The underlying numbers make the case for him: he scored 22 goals and provided eight assists last season as he fired Galatasaray to the Super Lig title, having completed a permanent €75m move from Napoli, a Turkish transfer record, after an initial loan.
Since arriving in Istanbul he has scored 59 goals in 74 appearances, with his physical presence and aerial threat drawing comparisons in Turkish media to former Chelsea forward Didier Drogba.
Arsenal have already missed out on two other attacking targets this summer, Morgan Rogers and Vinícius Júnior, and remain determined to add a marquee forward before the window closes. Softening Galatasaray's stance would not be unprecedented: Newcastle were similarly adamant that Bruno Guimarães would not be sold before a deal was struck earlier in the window.
Any move would likely be funded in part by departures. Both Martinelli and Nwaneri, who spent the second half of last season on loan at Marseille, have been made available, with Arsenal ready to sanction exits for the right offers.