The Toffees got their campaign off to the perfect start by beating Bosnich's former club Manchester United 1-0 at Goodison Park on Monday.
After the win, the Australian believes that the Merseysiders could challenge for a place in the Champions League.
"In Fellaini and [Nikica] Jelavic they have two fantastic players and they have got experience in Phil Neville, [Sylvain] Distin and Phil Jagielka. And the back four in general are really good, plus they have an experienced goalkeeper," Bosnich told talkSPORT.
"If I was an Everton supporter, I'd be looking at this season more than any other thinking why can't we [challenge the top teams]? If you looked at Liverpool, Tottenham and even Newcastle at the weekend, why can't they finish ahead of those teams and perhaps push for a fourth or fifth place?"
Everton travel to Villa Park on Saturday to face Aston Villa in their second game of the season.