The 34-year-old's current contract runs out in January, but the Scottish club want him to sign a new deal that will see him stay on.
Brown understands that Naysmith has struggled with injuries since his summer move in August, but wants him to be part of his squad for the whole of the 2012-13 campaign.
"We want Gary to stay," he is quoted as saying in The Daily Record. "He is an international defender with a lot of experience.
"He has been unfortunate with injuries but he is getting back to fitness and I'm sure he will be very influential in the coming weeks.
"We'll sit down with him and see if we can arrange a new deal but it will be up to him and his family. I know he really enjoys being at the club and is desperate to do his bit to help us.
"It won't be long until he's fit. It will be great to be able to call on a player of that calibre. Hopefully we will have him for longer."