According to Calciomercato (h/t Lucas Sposito of Sport Witness), the Premier League and La Liga leaders are both seeking to strengthen their options at centre-back in the midseason window and have identified the Italian as a possible target after his struggles at Milan.
Per the piece, Barcelona are poised to lose Javier Mascherano in January, leaving them with a lack of depth in central defence, especially with Samuel Umtiti injured.
City have John Stones sidelined too, while skipper Vincent Kompany has struggled to stay fit in recent seasons. With Eliaquim Mangala and Fernandinho filling in at centre-back at times this term, they would benefit from another option in this area.
Bonucci was regarded as one of the best defenders in the world before he made his move to Milan, having excelled at the hub of a formidable Juventus back line.
He had a torrid start to the campaign after changing clubs and trying to organise an unfamiliar set of players. However, as noted by David Amoyal of ESPN FC, there are small signs he is beginning to settle down into this new role at Milan:
Anyone who tracked Bonucci during his time in Turin and with the Italian national team will be aware of just how skilled the defender is.
At his best, typically in the middle of a back three, Bonucci's instincts for danger are sharp. Positionally he's exceptional, although when needed the Milan man can be aggressive too. His leadership skills and ability to organise team-mates also shone through at Juve.
As WhoScored.com noted at the end of last season, Bonucci is so often the man to get his team going:
If Barca or City were in a position to get this deal done in January it would be considered a massive coup.
Although both sides, often renowned for being soft defensively, have shored up this season at the back, a footballer of Bonucci's calibre would add much to the side, both on and off the pitch. As noted by Sposito, Guardiola is a known admirer of Bonucci, having tried to land him in his first summer as City boss.
As relayed by the Manchester Evening News Twitter account, the defender has spoken about the City chief lately:
It's been a tough week for Milan, so rumours about Bonucci potentially moving on are no surprise. As relayed by Paolo Bandini of the Guardian, there have been questions about the club's ownership by UEFA, speculation about Gianluigi Donnarumma's future and they were beaten 3-0 by struggling Verona in Serie A last time out.
Still, of all the signings the Rossonerri will be desperate to cling on to, Bonucci has to be the priority, as it was deemed such a shock to see Milan get him. A January departure feels unlikely, although if Milan don't secure any European football for next season, the summer is potentially another matter entirely.