The 27-year-old has played just twice for City's first team since March 2003 due to a series of injuries and recently the presence of England's number-two keeper David James.
But Weaver has finally returned to full fitness, playing 14 games on loan at Sheffield Wednesday earlier this season, and insists he will be pushing James for a place in City's starting line-up next season.
"People who think I have taken an easy option to sit on the bench for a year are severely mistaken," Weaver told the the City website.
"All professional footballers want to play and so do I - for Manchester City.
"I suppose I could have gone somewhere else and played first-team football, but I genuinely feel I am in good enough shape for the first time to put real pressure on Jamo for the starting role. I still believe I am young enough to have a future here."
If Weaver signs, it will be his 10th season at the club after signing from Mansfield in April 1997, and he has made 176 appearances for the Blues.