The winger only signed a new five-year contract at the Etihad Stadium in January but has grown frustrated after failing to hold down a regular place in City's star-studded side.
The Black Cats are thought to be keen on bringing the England international back to the North-East, while he has also been linked with Tottenham and Everton.
City boss Roberto Mancini understands Johnson's frustration but is reluctant to let players leave as his side prepare to juggle the defence of their Premier League crown with UEFA Champions League football.
However, the Italian is thought to be chasing Swansea's Scott Sinclair and Johnson's exit could hasten the arrival of the winger.
Mancini said: "Adam deserves to play because he is a good player, but it is not easy for me because I can only have 11 in my team. Every player who doesn't play is frustrated. I can understand that.
"I will talk to him about what will happen now, but it is difficult to let him go because I need to have all my players at this moment.
"When we start the Champions League, we hope to be playing a game every three days for 10 months, so it is not easy to decide about Adam."
Johnson has denied reports of a row with Mancini following City's Community Shield win over Chelsea last Sunday, but is keen to talk with the manager about his future after blaming his England snub at Euro 2012 on his lack of playing time at the club.
"I will be speaking to him about my situation," said Johnson. "There's been no bust-up. But I think what happened to me at City last season was why I missed out at the Euros."