They were negotiating for it from quite early on, and Lee was always very keen for City to make the move. I worked for a law firm in town in 1995/6 and we acted for a couple of the board members at City at the time. Colin Barlow* was definitely in talks with the Council at that point. IIRC, they showed an interest even at the initial stage before Manchester was confirmed as host of the Commonwealth Games - at that point, they were trying to have it named national stadium, ahead of Wembley (which was never seriously going to happen).
It was in September 1998 that the Council and Sport England agreed in principle for City to occupy the stadium, but the lease was signed in August 1999 (on the pitch**, before the opening home game of the season against Wolves). Gary James has written that Sport England would probably have cancelled the plans had we lost the play-off final to Gillingham, hosting the Games in a largely temporary venue that would have remained an athletics stadium afterwards.
* - If you were to argue that Colin Barlow definitely wasn't the calibre of CEO that City needed even in those days, you'd definitely find an ally in me.
** - It was a really windy day, and older readers might remember the stiff breeze lifting Howard Bernstein's combover vertically upwards as he signed.