The Eredivisie champions have swooped for the 22-year-old after he spent last season with AC Milan
The 22-year-old was tipped for big things after breaking through at Camp Nou as a teenager in 2007, but his career has faltered since then and he has spent the last two seasons on loan at AC Milan and Roma.
But that has not stopped the Eredivisie champions from stepping in and offering him another chance to shine and Overmars expects him to do just that.
He told the club's official website: "I'm very happy with the transfer. We wanted Bojan for the right wing, but he can play more positions up front.
"Bojan is 22 now, so it's time that he shakes of the label 'talent'. He's a team player and can score goals. That is what we expect from him."
The Dutch giants and the Blaugrana have enjoyed a positive relationship when it comes to transfers in recent times, after Isaac Cuenca spent last term on loan at the Amsterdam Arena.
And Overmars, who played for Barcelona between 2000 and 2004, believes the impact of himself and coach Frank de Boer - also formerly of Barca - was key in securing Bojan's signature.
"Ajax plays in the Champions League and he wants to play on the highest possible level," he added.
"Even when he was young, Ajax was a favourite club, I heard. This is an advantage that we have and of course it helps that Frank de Boer and I know a lot of people at Barcelona. The fact that the loan of Isaac Cuenca (last season) was handled well gave us a good image."
The forward revealed his regret at moving to Serie A two years ago after being unveiled as an Ajax player and admitted he may have been better off moving to the club sooner.
"My choices for Roma and AC Milan were wrong. Perhaps I should have already opted for the Eredivisie back in 2011," the attacker told De Telegraaf.
"I am a young player and I need to get regular first team action. I cannot make progress if I only get 18 games or so per season. I will get the chance to take the next step in my development at Ajax."