The utility back, who can operate at stand-off, centre and full-back, is currently out of action with the ankle injury which cost him his place in the Dark Blues’ RBS 6 Nations squad.
But in-coming boss Rennie has seen enough already to decide he wants the 27-year-old Fifer to be part of his squad once he arrives to replace Gregor Townsend this summer.
Rennie has already sanctioned new deals for Stuart Hogg, Henry Pyrgos, Tommy Seymour and Tim Swinson, as well as moves for Stormers pair Huw Jones and Oliver Kebble.
But Mark Bennett, Josh Strauss and Gordon Reid have been deemed surplus to requirements by the New Zealander - currently coaching Super Rugby franchise the Chiefs.
Horne joined Warriors in 2009 and has gone on to make more than 100 appearances for the Scotstoun outfit, helping Townsend’s team to the 2015 Guinness PRO12 title.
Horne, whose new deal will tie him to Glasgow until 2020, said: “I’ve loved my time here and I love the club. I’m passionate about playing my rugby in Glasgow.
“When the club came to me it was a no-brainer. We’re going places, which was summed up this year with us making the European Rugby Champions Cup quarter-finals for the first time.
“As soon as we signed up Dave Rennie for next season it made it an easy decision. When his name was mentioned it filled me with excitement as I’ve watched a lot of Super Rugby and the Chiefs are a great team. He plays a brilliant brand of rugby and it seems like all the players really play for him.
“I’m hungry to get better as a player and all the boys here are hungry to do better. I feel like I’ve played a big part over the last seven or eight years and it’s a journey I want to continue.”
Townsend added: “Peter is one of our hardest working players and epitomises what it means to be a Warrior, setting himself and others high standards in training every day.
“He deals with injuries and takes on new challenges with positive energy, which inspires everyone at the club.
“It’s great news that he’s staying in Glasgow for another three years.”