Press Association Sport understands the club will send a letter of complaint to senior figures at the Football Association and manager Manuel Pellegrini has threatened to field an under-strength team.
The game will be televised live by the BBC at 4pm on Sunday, February 21, after City had expressed a preference for the Saturday.
Quadruple-chasing City feel they are being penalised for their success with unnecessary fixture congestion during a critical 11-day spell in their campaign.
The tie will take place the day before Pellegrini's squad travel to Ukraine ahead of their Champions League game against Dynamo Kiev on February 24
That in turn comes just four days before the Capital One Cup final against Liverpool at Wembley, and City then face the Reds again at Anfield in the Barclays Premier League three days after that.
Senior sources at City feel the club are not being supported sufficiently by the authorities for respecting all the competitions - Pellegrini rarely fields weakened teams - and for representing the English game in Europe.
The problem in this instance arises because Chelsea's neighbours Fulham play in the Championship on Saturday, February 20 and police would be unwilling to sanction another kick-off close to that time
City thought a slot could have been found on Saturday but this was not possible.
A statement from the BBC read: "Despite continued efforts to schedule Manchester City's FA Cup fifth-round visit to Stamford Bridge on Saturday, conflicting fixtures and safety concerns meant this was not possible, so Sunday was the only slot available."
It is understood a Friday night game was considered but this would not have fitted with the BBC's schedule
That might also have created an issue with Chelsea, who have their own Champions League game at Paris St Germain the previous Tuesday.
The FA has not yet commented but it is understood all options were considered and, despite best efforts, a solution to the satisfaction of all parties could not be found on this occasion.
Source : PA
Source: PA