Striker Delap and winger Gittens missed Chelsea's 3-2 League Cup loss in the first leg of their semi-final against Arsenal on Wednesday, with Rosenior saying after the defeat that the pair had "really high temperatures".
"Liam (Delap) is still at home at the moment, as is Jamie Gittens. The club doctors are doing everything they can to contain what looks to be a virus because some members of our staff have gone down as well," Rosenior told reporters.
"We had a meeting today with the players about washing their hands, the basics, which is really important, and hopefully we can contain it. We have a busy period, so we need everyone ready to go and fit and healthy."
Chelsea, who have one win in nine Premier League games, will be desperate to get their campaign back on track when they host fifth-placed Brentford on Saturday.
They could be handed a major boost with the return of key players Reece James and Cole Palmer, who missed the FA Cup win at Charlton Athletic and the match against Arsenal after picking up knocks last week.
"Both of them are training today," Rosenior said.
"We just need to assess them after training and make sure they come through all the protocols they need to come through, but it would be great to have them back in the squad and at the moment, it looks like they could be."
The Englishman was also asked whether his first choice for goalkeeper had changed after Robert Sanchez committed two costly errors against Arsenal.
Sanchez failed to deal with Declan Rice's seventh-minute corner, allowing Ben White to stab in the opener. The goalkeeper then spilt White's cross early in the second half, leaving Viktor Gyokeres to tap home in the 49th minute.
"I am here to assess every position. I don't have number ones or number twos, that's not how I work," Rosenior said.
"If you look at me at every club, there has to be competition. Not just in the goalkeeping department but in every position on the pitch and you have to earn your spot in this team."