A summer switch is being mooted for the England international, who has slipped down the pecking order at Anfield.
Fierce competition for places on Merseyside, with Jurgen Klopp boasting a fearsome collection of attacking talent, has left Lallana filling a support role after a series of injury problems.
A similar standing would likely be taken up at the Etihad Stadium, given the abundance of riches on offer to Pep Guardiola, but the 31-year-old would add even more creativity to the Premier League champions.
Mills believes he would be a good fit for the treble-winning Blues, telling Football Insider: “You almost forget about him but he is very gifted and a talented footballer.
“I would almost say he more fits the Man City style, he’d be perfect for them with his short, darting runs and clever runs in behind.
“As you start to get older, you realise you are not invincible and your career is not going to go on forever.
“Liverpool will definitely want to keep him but it might come down to him wanting to reignite his career.
“If that’s the case, he’s going to go. He’s such a good player he won’t be short of interested clubs.”
There has been no suggestion that City would consider a move for Lallana were he to be made available, with most of the speculation at present focused on a potential return to Southampton.
Saints legend Francis Benali has already stated that he believes the former favourite would be welcomed back at St Mary’s despite forming part of a controversial exodus from the south coast.
He told the Daily Star of the man who made a £25 million ($32m) move to Anfield in the summer of 2014: “It’s a difficult situation for him to get back in the [Liverpool] team.
“It’s a scenario that many players would find difficult.
“But what would be the tough thing is would the supporters welcome players [like Lallana] back to the club?
“It’s a relationship between the two clubs that has certainly spiked the nerves of Southampton fans with a lot of our top players leaving for Liverpool.
“I just like to see good, talented and passionate players that want to play for Southampton - playing for my team.
“However that happens, I’m quite open to anything.”