The German born USA international played in all 15 of Rovers' fixtures since his January arrival and helped Kean’s men stave off the threat of relegation on the final day of the campaign at Wolves.
The former Eintracht Frankfurt and Bayer Leverkusen player is said to have settled into English life well according to the Scot, who wants to sit down with Schalke officials and tie up a deal over the next fortnight.
"If we can keep him at the club that will be fantastic," Kean said. "Jermaine is happy, his kids are in school.
“Hopefully he can see what we have done and we can get him on the books.
"We will sit down with Schalke in the next couple of weeks as he has been outstanding for us."
Now under their first summer of Venky's Group ownership, it is expected Blackburn will be active in the coming transfer window as they look to build on Premier League survival, while hoping to hold onto young talents including Junior Hoilett and Phil Jones.