The Wales international, like Georgios Samaras, is in the last few months of his contract with their respective futures subject to much speculation.
Ledley has been linked with Crystal Palace while Hull City are reportedly interested in the Greece forward.
Ledley scored the only goal in the home Scottish Premiership clash with Partick Thistle on New Year’s Day, with his drive from 25 yards extending Celtic’s unbeaten league run to 19 games.
Afterwards, manager Neil Lennon revealed the midfielder had “intimated that he would like to stay,” a view with which Brown concurred.
“We all want Joe to stay,” said the Scotland midfielder. “The gaffer and everyone at the club wants him to stay so, hopefully, that can happen.
“It’s a boost for the squad to hear that he wants to stay. Joe is huge for us. You could see that against Thistle. His performance was outstanding, as was the goal.
“We know him and Sammy are heading towards the end of their contracts, so it would be great to keep them.
“Joe is up there with the best midfielders I’ve played with. He scores important goals and [the goal against Thistle] was another huge one for us to keep up the unbeaten record.
“It’s important to keep a core of players – especially through the middle of the team.
“We lost Victor Wanyama and Gary Hooper in the summer but, hopefully, we can keep Joe. It would be huge for the club and huge for me.”
“We had a conversation on Tuesday,” said Lennon.
“He has intimated that he would like to stay and we are hoping that we can speak to his representatives this week and get that finalised.
“You are never confident, but hopeful.”