According to Sami Mokbel of the Daily Mail, the two Premier League sides have "concrete interest" in the playmaker after a move to Barcelona failed to materialise in the summer. Paris Saint-Germain are also said to be keen on Seri, while Arsenal were linked ahead of the season.
Mokbel added that Seri is "open to a move to the Premier League" and that "scouts from City and Liverpool have watched Seri extensively this season and will continue to keep a watching brief ahead of a potential swoop."
Seri enjoyed a sensational season with Nice last term, as the team finished in third place in Ligue 1 and earned a place in the UEFA Champions League qualifying round.
These numbers from Squawka Football sum up just how crucial he was for Lucien Favre's side, offering a controlling and creative presence in the middle of the field:
Interest from Barcelona didn't come as a massive surprise, especially with the player available for a fee of £32.5 million, as noted by Mokbel.
The Ivory Coast international is an extremely gifted footballer. Seri is always seeking to get on the ball, can glide away from onrushing opponents and constantly keeps opposition defences on their toes due to his passing accuracy and range.
This campaign hasn't gone so well for Seri or for Nice, though. The midfielder has spent a month on the sidelines due to injury, and after performing so well last term, the side are down in 14th in the table, just four points clear of the relegation zone after 16 games.
As a result, sides like City and Liverpool may view the January window as an opportune time to make a play for Seri. But as relayed by Get French Football News, the midfielder made it clear earlier in the campaign he will not be jumping ship midseason:
It's tough to see exactly where Seri would fit into either Liverpool's or City's midfield anyway.
The Reds have an array of options in the centre of the pitch to call upon, with Philippe Coutinho, Adam Lallana, Emre Can, James Milner, Georginio Wijnaldum and Jordan Henderson all vying for places. Meanwhile, City boast the two standout midfielders in the division in Kevin De Bruyne and David Silva, with Ilkay Gundogan an excellent deputy.
One side that may benefit from an addition in midfield at this point is PSG, as ESPN FC's Jonathan Johnson noted after their defeat to Bayern Munich on Tuesday:
Jonathan Johnson @Jon_LeGossip
With Unai Emery unwilling to give youth academy talents a chance, preferring to play some players out of position or reconvert them & forcing other settled midfielders to play DM instead, PSG’s options in the middle are essentially threadbare. New DM a must in January.
In terms of style, Seri is not someone that'll sit deep and protect the defence, but he'd bring composure and depth to this area of the field.
If he is to leave Nice, the Ivorian would want to move to a club where he'd remain a key man and where there's a clear role for him in the team. At the moment, neither Liverpool nor City feel as though they'd be able to offer him those kinds of assurances.