The 24-year-old was set to sign a new deal after being offered the captaincy by the new management team of John Barnes and Jason McAteer.
But, after the disappointment of missing out on the League One play-offs on the final day of the season, Jennings opted for a fresh start by agreeing a one-year contract at Fir Park.
"I was going in on Tuesday morning to sign and got a phone call on Tuesday morning," Jennings said. "The offer was too good to turn down in the end. I'm gutted to be leaving Tranmere, I was brought up there.
"But the chance to play in the SPL, personally I think it's a great opportunity for me. I'm looking forward to getting my head down for 12 months.
"However, if I'm enjoying it and they offer me an extended contract, I'll stay. If I'm not enjoying it, I'll keep my options open."