The two sides meet at St Mary's Stadium on Sunday, with Chelsea leading the Premier League and Southampton back on track in fourth place.
Koeman's side have overcome a mini-wobble that saw them six games without a win in all competitions during November and December, with the Dutchman renewing acquaintances with Mourinho this weekend.
The pair worked together on Louis van Gaal's coaching staff at Barcelona in the late '90s and Mourinho revealed he always felt Koeman was the right man to take over from Mauricio Pochettino.
"I know Ronald well, we worked together. We worked together in Barcelona on Mr Van Gaal's coaching staff so we were colleagues," said the Portuguese.
"I know that he's a good coach and has a good mentality and I know the football principles he defends and he likes.
"So for me, when I saw him get his hands on a Pochettino team, I thought he was the right guy to do it."
Mourinho also praised Koeman's ability to get the best out of a squad that lost key players such as Adam Lallana, Rickie Lambert, Luke Shaw and Dejan Lovren in the close-season.
"Obviously they lost some good players but they were very clinical in their choices," Mourinho added.
"You lose Lambert, you go for [Graziano] Pelle. This kind of situation, you lose Lovren, you keep [Jose] Fonte and you go for [Toby] Alderweireld. You lose [Artur] Boruc, you go to the big guy [Fraser Forster].
"They have a very good squad, a very good coach, they're playing well and they're getting results. They don't have pressure, they're the kind of team that in this moment has no pressure because relegation is very, very far [away].
"The title is not for them, so they are in a fantastic position to play beautiful football like they are doing."
