The former Liverpool manager believes under Koeman the Toffees will have what it takes to gatecrash the top six.
Koeman replaced Roberto Martinez as Toffees boss this summer, a move which has bred new optimism on the blue half of Merseyside after back-to-back bottom-half finishes.
The Dutch boss led Southampton to sixth place last term, the Saints' highest finish since 1985, a placing which qualified the south coast club for their first ever European group stage campaign.
There are hopes amongst the Goodison Park faithful that he can have a similarly impact at Everton and restore them to their status as challengers for European qualification.
Hamann, best known for his seven-year spell with the Toffees' cross-city rivals Liverpool, certainly seems confident that Koeman can bring the good times back.
He believes this year it can be Everton who follow the lead of sides such as the Saints, Leicester City, and West Ham United in upsetting the traditional balance of the upper echelons of the Premier League.
Hamann told the Racing Post, via the Liverpool Echo: "We saw Leicester - and to a lesser extent West Ham and Southampton - break the stranglehold of the big guns last season and this term I can see Everton being the best of the rest and maybe nicking a top six spot.
"I’m a huge fan of Ross Barkley who I know seems to divide opinions at Goodison, but he’s a matchwinner. I’m sure John Stones is only going to improve as a defender, especially under Ronald Koeman who is the outstanding manager the club needs."
Much of Everton's success this term will depend on them keeping the England international pair of Barkley and Stones, and hanging on to top goalscorer and prize asset Romelu Lukaku.
The Toffees kick off their season in nine days' time with a home clash against Tottenham Hotspur.