Reports suggested Arsenal officials planned to use Tuesday's trip to St Mary's to open negotiations over the 17-year-old left-back.
But asked if Gunners counterpart Arsene Wenger had mentioned anything about Shaw after the game, Adkins said: "No, he didn't say anything.
"As I have said before, Luke is better playing for Southampton Football Club, where he is being nurtured and looked after.
"He is still only 17 and playing in the Premier League, where he is getting an opportunity to flourish, to make some mistakes and to keep kicking on.
"We have a good culture here. He is under contract, we are not looking to let him go.
"This is best place for him at this point in his career. He has a big, big future in the game, which he demonstrated with his performance against Arsenal the other night.
"If anyone wants to try and do business, all the best dealing with the chairman."