The Dutchman has led Saints to second place in the Barclays Premier League after replacing Mauricio Pochettino over the summer, with a number of his recruits already starting to make names for themselves in the division.
Graziano Pelle was brought in from Koeman's former club Feyernoord and, as well as scoring on his Italy debut, he has notched six league goals and picked up the player of the month award for September.
Dusan Tadic has arguably had an even better start, with the Serbia winger up for the October award having starred in the 8-0 demolition of Sunderland.
Leicester are the visitors to the south coast on Saturday before a tough run of fixtures that will see Koeman's side face Man City, Arsenal and Manchester United in the space of eight days.
Reigning champions City are currently third in the table with runaway leaders Chelsea looking almost unstoppable, but Koeman believes his Southampton charges could deal blows to either side's title hopes, even if Saints cannot last the pace to challenge for the crown.
"I think it is a two-team race because I still believe that City and Chelsea are stronger over the whole season," he said.
"In one game I think we can beat them but over the whole season normally they are much stronger than us.
"We know our situation and we know our ambitions but we live game by game, and it is not important and not necessary to look more forward because, in my opinion, that is not the best way to be successful.
"Of course we would like to continue in the good direction the club has been going in the last three or four years
The next step is playing in Europe, we know that.
"It is a dream for the club and a dream for the fans
We have to do it but I have confidence that we can reach that."
The Foxes travel to St Mary's having lost four of their last five matches and without a win since their famous 5-3 victory over Man United.
But, with a stern test to come in the run-in to Christmas, Koeman believes Nigel Pearson's men will still prove a threat.
"We know our schedule for the month of December and the end of November and we will have a tough week at the start of December, but we look forward to that," he said.
"It gives us the possibilities to play well and see what we can do against the big ones, but the first one we have to play is this weekend against Leicester and maybe that is more difficult than playing Man City."
Other than long-term injury absentee Jay Rodriguez, Koeman has a fully-fit squad at his disposal for the weekend and the former Barcelona midfielder believes current form, which has seen Southampton win 10 of their last 11 fixtures in all competitions, is key as no player wants to miss out on such a run.
"The team and the players are still in good shape, we don't have injuries
The team is fit and full of confidence," said Koeman.
"We had a good pre-season, that is important, and we have time between the weekends to do training sessions because we don't play European football.
"We are winning, that is the most important motivation to be in a good shape in a physical way because it is easier to keep going."
Source : PA
Source: PA