Mills comes out of contract at the end of the current campaign and has already indicated that he won't be making any decisions on his future until the Spurs' season has concluded.
The 25-year-old is likely to receive tempting offers to join a rival franchise, but Gaze would like to see him stay in San Antonio and continue his career under the watchful eye of veteran coach Gregg Popovich.
"I am hopeful that he (Mills) can find a way to stay with the Spurs," Gaze, who won an NBA Championship at San Antonio under Popovich in 1999, said.
"The Spurs have meant a lot to Patty and they have been really patient with him for a period with his development.
"He isn't in a program that is looking for a quick fix and he is playing well which indicates that he is very comfortable."
Mills is currently having his best season in the NBA, having increased his output to average over 10 points a game while receiving more than 18 minutes of court time behind veteran point-guard Tony Parker.
It that kind of form that has caught the eye of rival clubs and has Mills facing a huge dilemma at the end of the season when he weighs up his future.
"I am not sure of the exact technicalities of his contract, but he has options to consider in the off-season," Gaze added.
"I would like think that he has found himself an unbelievable organisation with one of the all-time great coaches, but having said that I don't think anyone is going to begrudge him if someone comes out with a godfather offer to at least have the right to consider those other offers if they are on the table."
Gaze said he has been impressed with Mills' performances this campaign and a lot of that success could be attributed to the fitness work he did during the off-season.
"He has shown glimpses over the last couple of years, he just hasn’t been able to establish himself in the rotation," Gaze said.
"This year he is well and truly entrenched and he is getting quality minutes and putting up some massive numbers.
"He has always been in shape, but now he has just got himself into that elite level shape.
"A lot of his success can be attributed to conditioning and the shape he has got himself into and it is full credit to him."