According to the Telegraph, the Potters are ready to offer around £12m (€13m, $15.5m) for the former Aston Villa midfielder, as Mark Hughes looks to add an eighth summer signing to his squad.
Stoke are unlikely to be able to match Delph's current wages, but the club are keen to get the deal done and have not ruled out the option of signing the England international on loan.
Delph moved to City in the summer of 2015 for £8m after six years at Villa, but opportunities have been at a premium for him at the Etihad and the 27-year-old has made just 40 appearances in all competitions since swapping the Midlands for Manchester.
While Bernardo Silva remains the only midfielder brought in by Pep Guardiola this summer, Delph is firmly behind David Silva, Fernandinho, Yaya Toure and Kevin De Bruyne in the pecking order.
According to the Daily Mail, the Bradford-born midfielder, who has featured in just 10 Premier League games over the last two seasons, has also attracted interest from Leicester City but he is understood he would prefer linking up with Hughes at the Bet365 Stadium.
Delph's arrival would pave the way for Giannelli Imbula to leave, amid interest from a number of Ligue 1 clubs and Serie A outfit Torino. The French international, who is yet to feature this season, arrived for club-record fee of £18m in February 2016 but has failed to live up to expectations, featuring in just 26 league games in two seasons.
Meanwhile, Stoke have confirmed the signing of Kevin Wimmer on a five-year deal from Tottenham Hotspur for £18m. Crystal Palace, Southampton, West Ham United and a host of clubs from the Bundesliga and Serie A were also said to hold an interest in Wimmer, who only made 31 total appearances for Tottenham.
However, the Austrian international opted to move to the Potteries after being impressed by Hughes' plans. "I had some really good conversations with the manager, Mark Hughes, and they were important for me," he told the club's official website. "He is a very good manager and I like the way he approaches the game, his style of play suits the way I like to play too - he likes his defenders to be able to play out with the ball.