According to Sky's Florian Plettenberg, the Turkish champions have formally submitted their offer for the 25-year-old winger, who has become Galatasaray's top transfer priority as they look to strengthen their attack ahead of a Champions League campaign.
Martinelli, who joined Arsenal from Ituano as a teenager in 2019 and has since made 278 appearances, scoring 62 goals and winning last season's Premier League title, is contracted until 2027, and Arsenal are understood to value him closer to £50 million, meaning the current bid falls some way short of what would be required to force a deal through.
Manager Mikel Arteta and sporting director Andrea Berta are also reluctant to sanction a sale before bringing in a replacement on the left flank, having already lost Leandro Trossard to Besiktas earlier in the window.
Arsenal's other significant move this week concerns their own defence rather than a departure. The Premier League champions are reigniting their interest in Aston Villa central defender Ezri Konsa, who remains on holiday following the World Cup and was absent from the squad for Wednesday's Super Cup defeat by PSG.
Having already signed Bruno Guimaraes from Newcastle, Arsenal have turned their attention to reinforcing at the back, with Konsa's versatility seen as a particular asset at the Emirates.
An initial offer of around £30 million for the 28-year-old did not progress over the summer, with Villa valuing Konsa at closer to £60 million after a season in which he made 48 appearances in all competitions and won the Europa League. Arsenal accept they will need to significantly improve on that opening bid if they are to succeed, though whether Villa are prepared to sell below their own valuation remains uncertain.
Sources have also told the BBC that Arsenal have held exploratory talks with Tottenham over a possible move for club captain Cristian Romero, even though Spurs are believed to have no intention of selling to their fiercest rivals. The 28-year-old was ruled out for the remainder of last season by a knee injury sustained against Sunderland in April, but recovered in time to start for Argentina at the World Cup, where he played 70 minutes of the final defeat by Spain.
Given the intensity of the rivalry between the two clubs, any deal for Romero would rank among the most surprising transfers of the summer, though Arsenal's willingness to make enquiries at all underlines the scale of their defensive rebuild following William Saliba's injury.