According to Mail Sport, fellow Premier League club, Everton, were in talks to land the 23-year-old, before the Gunners stepped in on Monday.
It is understood that Italian giants AC Milan, now taken over by Chinese ownership also wanted Kolasinac, but the Bosnian has his heart set on working with Arsene Wenger.
He will join up with Arsenal officially in July.
His reps have reportedly been told that the Frenchman will not leave the Emirates at the end of this season.
Wenger sees him as competition for Nacho Monreal and Kieran Gibbs.
The young left back, who has made 18 appearances for Bosnia, was the subject of a failed £5.3million bid from Chelsea in January.