The former Rangers, Tottenham and Aston Villa full-back is currently without a club.
Hutton, now 34, has been capped 50 times for Scotland and has played with Forrest at international level.
"Meeting up in the Scotland squad and things, he’s a really nice guy, down-to-earth. He’s one that’s got everything," Hutton told Football Insider.
"I just see him maybe flourishing at a team like Leicester or maybe one of these type of teams – obviously a manager that knows him well.
"I really think he’s got everything that it takes to make it down in the big league."
Brendan Rodgers is the manager Hutton refers to, who knows Forrest from his time at Celtic.
Forrest has seven goals and five assists in 13 games in all competitions already this season.
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