Delph, 29, turned out 20 times for the Blues in a season plagued by inconsistency.
Inter Milan were rumoured to be keen on a move for Danilo but negotiations over a deal have since broken down, while Delph has been linked with a move to boyhood club Leeds United.
Should the pair leave, City boss Pep Guardiola will be left with a shortage of cover in the full-back options, with Aleks Zinchenko, Kyle Walker and Benjamin Mendy the sole full-back specialists on the books at the Etihad.
Here, MEN Sport looks at possible transfer targets for the City boss – with a particular focus on right-backs.
The Crystal Palace right-back, 21, has earned rave reviews for his opening two campaigns in the first team at Selhurst Park and is regarded as England’s brightest prospect in the position.
Guardiola is believed to be an admirer of Wan-Bissaka, whose pace and power rival those of City’s first-choice right-back, Kyle Walker.
The Portugal international, 25, moved to Juventus from Valencia last summer and made 25 appearances in Serie A. Cancelo, who has also played for Inter Milan and Benfica, is believed to be viewed as dispensable at Juventus and, like Wan-Bissaka, possesses the pace required to operate at full-back in a Guardiola side.
The Chelsea full-back, 19, spent last season at Wigan in the Championship and proved himself as one of the most impressive full-backs in the division. James, who was nominated for the EFL Team of the Season, is expected to break into Chelsea’s first-team next season.
The Derby right-back, 18, made 40 appearances for the Championship play-off finalists last season and, like James, caught the eye with the consistency of his performances. Speed is a particularly notable attribute.
The Southampton right-back, 27, spent the second half of last season loan at Inter Milan, who have the option to buy the 33-times-capped Portugal international this summer. Soares’ contract at Southampton expires next summer. He could represent a cheap alternative as City hunt a back-up full-back.