The Bosnian was made aware that he was behind Kieran Tierney in the pecking order after lengthy talks with Mikel Arteta.
Former club Schalke had been keen on signing the left-back on loan but Arsenal would rather recoup a fee for the defender.
West Ham expressed an interest last week and had been prepared to put up a fee for the 27-year-old, who joined the club on a free transfer from Schalke in 2017.
However, according to the Guardian, Kolasinac is not interested in a move to the London Stadium and instead wants to fight for his spot in north London.
Tierney is Arteta’s first choice at left-back following the Scot’s impressive displays since the return of the Premier League in July.
Kolasinac is aware that he will only provide back-up for the former Celtic ace but he believes that he can provide enough competition to convince Arteta to keep him at the club.
The Gunners have signed Pablo Mari, Willian and Gabriel so far this summer, while Pierre-Emerick Aubameyang is set to sign a new three-year deal.
A host of players such as Matteo Guendouzi, Mesut Ozil, Lucas Torreira, Sokratis and Shkodran Mustafi are all up for sale and if a number of those can be sold then Kolasinac may be given another chance to place his claim.