Koeman has been strongly linked with the Everton job.
Southampton manager Ronald Koeman
Momentum has started to gather around the Dutchman possibly taking the helm at Goodison Park but the Saints are said to be 'relaxed' about reports suggesting he will make the switch to Merseyside.
Everton players during the lap of honour
Bookies slashed odds on Koeman to succeed Roberto Martinez the next Everton boss as he was installed as odds-on favourite in many places.
This all comes at a time when the Daily Echo reports that Southampton are currently in the process of negotiating a contract extension with Koeman.
Southampton manager Ronald Koeman acknowledges fans
There is also mention that there has been no official approach for the Dutchman, and a lot could depend on just how much he thinks he would achieve at St Mary's.
Everton are certainly interested in Koeman, that is of little doubt and there is plenty of noise to suggest that they will do what they can to attract him, including a vast transfer budget and pay packet.
It does, however, seem pretty early on in proceedings suggest that Koeman is either close to leaving Southampton or being appointed the new Everton boss. There is some way to go, it seems, before there is firm movement in Koeman making the switch to Merseyside.
Everton investor Farhad Moshiri and chairman Bill Kenwright in the stands