Claudio Ranieri's Foxes sit three points clear on top of the Premier League table after 24 matches with a dynamic style of game which has won over new fans and created an enormous buzz around the country.
England's all-time appearance record holder Shilton, won a host of trophies with Forest including the the 1977/78 First Division title and back-to-back European Cups in '79 and '80, started his career with Leicester back in 1966 and can see similarities between the two teams.
"I follow all my former clubs but Leicester is my club. I stood on the terraces at Filbert Street as a kid," the 66-year-old told Sky Sports.
"Then to play for them as a teenager was something special. I think what they are doing now is unbelievable. There is no other word for it really."
He added: "I do see parallels. A year ago, I remember doing an interview about Leicester and they were seven points adrift of second bottom and everyone thought they were going straight back down from the Premier League. But they've turned it round.
"They've got a great spine to the team, which is what we had at Forest.
"Kasper Schmeichel has come on a lot and is starting to dominate in goal, just like his Dad did. The two centre halves [Robert Huth and Wes Morgan] are big imposing figures who don't mess around and know how to put a tackle in, just like we had with Larry Lloyd and Kenny Burns. They don't lose many games and it gives them a really solid base.
"Then of course there's the striker. We never had just one like that for us, but what Jamie Vardy is doing is incredible. He's just an out-and-out goalscorer."