The Senegal international put pen to paper on a four-year deal after moving from Salzburg during the transfer window.
He is yet to feature for the south-coast, however, after awaiting confirmation that a work permit would be issued.
Koeman confirmed that the paperwork had been completed ahead of Saturday's Premier League clash at Swansea City, but Mane will not feature at the Liberty Stadium.
"Yesterday I got a message that everything is done and the work permit is OK," he said.
"He is flying back today and will arrive tomorrow very early but it's too close for Saturday.
"Tomorrow I'll speak to him about his physical state because I don't know what he did in the last 10 days, but it will be a good chance for him to be part of the squad for the League Cup game against Arsenal [next Tuesday].
"He's back now and we have to wait until tomorrow, then he's part of the group."
Koeman also revealed that Shane Long would be fit for the trip to south Wales, after suffering a blow to the head in last weekend's 4-0 thumping of Newcastle United.
"Shane Long is okay, he's already training and has been doing sessions for three days so he's no problem for this Saturday," he said.
Dutchman Koeman has also praised Morgan Schneiderlin after the midfielder scored three goals in Southampton's first four Premier League games, having stated his desire to leave the club during the transfer window.
"I wasn't surprised how our fans supported Morgan because we have great fans and Morgan did a lot and is doing a lot for the football club," Koeman said.
"I said several times it was a difficult situation for him but he has a contract with the club and he's doing that in a very professional way and now you see he's happy to be part of the team."
