Osimhen currently earns more than €20m net per year in Turkey, a figure no European club can currently match, and Arsenal are determined not to overreach financially to land him, even though the reigning Premier League champions remain keen.
The 27-year-old, who scored 22 goals and provided eight assists in 33 appearances across all competitions last season, has 35 goals in 52 caps for Nigeria and is a profile Mikel Arteta admires, though talks with Galatasaray remain complex. He has already shown his sharpness this season, scoring twice in the Süper Lig's opening weekend.
With the transfer window closing in a matter of days, the decision now largely rests with Osimhen himself: stay in Istanbul on a lucrative deal with Champions League football guaranteed, or take on a new challenge in a league he has never played in, joining last season's Champions League runners-up and adding to his trophy collection in the process.
The Osimhen situation forms part of a broader search for attacking reinforcement at Arsenal, who want another forward option to compete with Viktor Gyökeres before the window shuts.
Atletico Madrid have accepted an offer from Arsenal for Julián Álvarez, according to Mundo Deportivo, though the Argentine has given his preference to joining Barcelona instead and has no intention of moving to the Premier League.
With that route effectively closed off, Arsenal are examining other options, of which Osimhen remains the most advanced, even if the financial calculation continues to give the club pause.
Arsenal's talks with Galatasaray over Osimhen developed out of separate discussions concerning Gabriel Martinelli and Ethan Nwaneri, both of whom have been linked with moves away from the Emirates Stadium this summer.
Osimhen joined Galatasaray permanently last year after an initial loan spell from Napoli, and has become the club's talisman since, helping them retain the Süper Lig title.
Should Arsenal decide to press ahead, the fee itself is understood to be more straightforward than the wage negotiations, with Galatasaray privately expecting an offer in excess of £60m to even begin serious talks.