The alleged chanting occurred during the second half of the third-round tie at the Etihad Stadium, which City won 4-0.
It comes two days after the FA said it was also looking into chanting of a similar nature during Manchester United's defeat of Everton on Friday.
Planet Sport understands the governing body will be liaising with the club and relevant authorities over the matter.
An FA spokesperson said: "We strongly condemn the use of the term 'rent boy' and we are determined to drive it out of our game.
"We continue to work closely with the Crown Prosecution Service, as well as the UK Football Policing Unit, in relation to the use of this term.
"Part of our work in this area has been to provide the relevant authorities with impact statements from LGBTQ+ supporters, detailing how chants of this nature affects their experience and feeling of inclusion at football matches, so that a clearer stance and understanding on the chant can be established.
"We stand firmly against all forms of discrimination and we are striving to ensure our game is a safe environment for all, which truly embraces diversity and challenges hateful conduct both on and off the pitch."
City said in a statement: "Manchester City strongly condemns the behaviour of a minority of fans who engaged in discriminatory chanting at today's match.
"We are proud to celebrate inclusivity in football and ask all fans to join us in creating a positive environment and atmosphere, where everyone is welcomed, accepted and has a fantastic matchday experience."
There also appeared to be some disturbances involving Chelsea fans late in the game, with police visibly stepping up their presence at the end where the visiting supporters were situated.