According to a report by Express, the Englishman's paperwork remains stuck in process.
"We're obviously hoping we get support from the city on that front. We are getting it at the moment, but naturally things need to move faster than they would for someone who doesn't work in football," said head coach Rene Wagner.
Heskey, the son of former England striker Emile Heskey, joined Cologne for a reported €2million fee. The fee could rise to as much as €8million through add-ons. Manchester City are also said to have secured a sell-on percentage, a buy-back clause and a right of first refusal as part of the deal.
The teenager becomes the second Manchester City academy graduate to join Koln this summer, following winger Jahmai Simpson-Pusey, whose loan spell at the club last season was converted into a permanent transfer.
Heskey progressed through every age group at City's academy after joining as an eight-year-old, and capped his development with a winning goal in the FA Youth Cup final against Manchester United before completing his switch to the Bundesliga.
Koln's squad planners moved quickly to beat several rival clubs to Heskey's signature, with technical staff at the German club well aware of his profile from previous scouting work in English youth football.
The delay over his work permit is not considered unusual for a player of his age and status, but Wagner's comments reflect a growing impatience at the club to have their new arrival available for selection as pre-season progresses.
Heskey made his senior debut for City in last season's Carabao Cup and was a prolific performer at every level of the club's youth set-up, contributing heavily to goals and assists across his time with the Under-18s and Elite Development Squad.
His departure continues a pattern of City moving academy graduates on to clubs willing to offer regular first-team opportunities, while retaining financial interest in their future value through sell-on and buy-back clauses.