The Dutchman guided Southampton to top-seven finishes in both of his seasons at St Mary's and makes his first return to the Hampshire club on Sunday after moving to Goodison Park in the summer.
Everton currently sit seventh but have stuttered after a flying start, with a 2-0 victory over West Ham their only win in eight matches.
The gap between the Toffees and the best, meanwhile, was starkly illustrated by a 5-0 defeat against Chelsea that could have been considerably worse.
Koeman said: "I'm really, really happy that I made the move to Everton because I think it's a big club
We know it's a project, it's not an easy one, but okay we are totally confident that we can reach finally what we'd like to do with Everton
"We started here and we had our opinions about how we'd like to play
Of course it's not that everything is okay because we have things to change, we need to change
"We like to play a little bit differently than maybe Everton did in the last couple of years
One time it's going well, the next time it's not how I like to see the team, but that's football
"You cannot change everything in two months
We need more time and that's difficult because you don't get time in football."
Everton have had a reputation for showing patience with managers but Koeman's predecessor Roberto Martinez quickly felt the heat from fans when his bright start turned sour
The club was taken over by Iranian billionaire Farhad Moshiri in February, meaning extra funds for the new manager but also increased expectations
"I don't ask for time because I'm realistic," said Koeman, who finally has a virtually fully-fit squad to choose from.
"Of course we try to do things differently and that needs time
It's not easy to change a system of playing, to change players or to expect different things of players, that is working every day
"One day it looks okay, it's what I want, then the first half against Swansea was really disappointing
The second half wasn't, but we need to be more regular during the whole game and we need to start much better than we did in most of the games."
Everton recovered from 1-0 down to grab a late draw against Swansea last weekend but Koeman knows they must do better, starting at Southampton on Sunday.
The opening of the transfer window in just over a month's time will give Everton the chance to strengthen, and Koeman has already expressed an interest in out-of-favour Manchester United winger Memphis Depay.
The Toffees boss said: "More players are linked to Everton than to other teams
Of course we're working to get the best players in and what we need, I think that's the most important.
"That's a situation maybe for January, finally it will be next summer
But now we are playing and of course my job is to get everybody prepared for the weekend and finally to have a team that I like and what I think can be the strongest team to achieve what we'd like to achieve."
Source : PA