Besic made his debut for Everton in Osman's testimonial match against Porto on Sunday, playing 45 minutes in the 1-1 draw in Merseyside.
And Osman said the 21-year-old Bosnia-Herzegovina international midfielder has the potential to become an Everton hero following his arrival from Hungarian outfit Ferencvaros on a five-year deal.
"He hasn't tackled that aggressively in training, thankfully," the 33-year-old told reporters.
"But the touches and quality he has shown in training.
"It was a really good debut for the lad and if he plays like that every week then he will be a crowd favourite for many years."
Everton kick-off their Premier League campaign at newly promoted Leicester City on August 16.