The 26-year-old full-back moved to Aberdeen in January and will be a free agent in the summer.
Logan made his Dons debut in the 4-0 League Cup semi-final win over St Johnstone and his dream loan spell has seen him pick up a winner's medal in that tournament.
The Brentford player has yet to play in a losing Dons side, and McInnes is keen to persuade the pacy defender not to join previous loanee Michael Hector in returning south..
McInnes said: "He brings real pace, good energy and he's very good defensively; one-versus-one.
"If Shaleum wants to be here next season, I think there's a good chance we could try and make that happen.
"He's played a lot of big games since he's been here. He's seen exactly what Aberdeen's all about."
