Bonucci arrived in Milan to seal his shock €40 million move from Juventus on Friday.
City came close to signing the Italy international last year, only for him to decide to stay at Juve because his son, Matteo, had fallen seriously ill.
The Blues had looked into the possibility of signing the 30-year-old towards the end of the 2016-17 season but did not believe the Turin club would sell.
Having recently been given encouragement that Juve were indeed open to offers, and for an attainable price, City made an approach to the player in the middle of June.
City had been more than willing to match Juve's asking price - with insiders claiming it would have be comparatively "easy" to agree a fee - but had struggled to agree personal terms with Bonucci.
Sources have told Goal that Bonucci was requesting considerably more from City and other overseas suitors than the reported €7m annual salary he will receive at Milan, as he weighed up how best to take care of his family.
The Italian was determined to leave Juve because his relationship with coach Massimiliano Allegri had completely broken down over recent months, but he was ultimately not prepared to move too far from Turin.
That ensured he was able to wrap up a move to Milan relatively quickly, though it leaves City looking elsewhere for a centre-back for the second time this summer.
Pep Guardiola has prioritised the signings of four full-backs and Alexis Sanchez, following the arrivals of Bernardo Silva and Ederson, but has also made it clear that he would like a new centre-back if the possibility arises.
Bonucci had not been a target for City earlier this summer but after moving on from Virgil van Dijk there was hope they could strike a deal.
It is understood the Blues still have two more viable options to turn to, including Real Sociedad defender Inigo Martinez.
Martinez was regarded as the third and least likely option while City pursued Bonucci and it remains to be seen how he will be regarded now the Italian has moved elsewhere.
City ideally want a strong, experienced defender who take the burden off Vincent Kompany's shoulders and also help mentor John Stones.
Guardiola felt Kompany's injuries, and the failure to sign Bonucci last summer, did not help Stones' development in a difficult first season at the Etihad Stadium.
On Friday the Blues signed Kyle Walker for a fee of £45m, potentially rising to £50m, and are trying to push through a deal for Mendy as soon as possible.
Negotiations are thought to be in the latter stages but Monaco, perhaps sensing a hint of desperation in City after they were snubbed by Dani Alves, are now holding out for nearly £50m, having previously wanted £40m.
But while City are well placed for many of their targets, it has been a different story in their pursuit of a centre-back.
Under the impression that Bonucci was not for sale, City had previously made Southampton's Van Dijk their top target for the position.
At that stage, City had not fully committed to the purchase of a new centre-back and would only make a move depending on several other factors, such as progress made with their priority signings and if they could sell enough of their current squad.
By the time they had decided to press ahead and sign a central defender, Van Dijk had already made it known that he favoured a move to Liverpool. That move is currently on ice, but City were put off when Southampton subsequently upped their asking price to £60m.
Sources have also told Goal that City would sell Nicolas Otamendi if they can bring in a new defender this summer.
The Argentine is seen as disposable by Guardiola, and it is hoped he would also fetch at least £20m in the current market.
Otamendi has not been in training with his team-mates this week, with City yet to clear up whether he is suffering an injury or not.
City hope to raise £100m through sales without taking Otamendi into consideration, though they have struggled to offload some of their unwanted players.
They had hoped to receive £20m for Joe Hart but have settled on a loan deal with an up-front fee as a compromise. That deal is expected to be announced over the weekend.