Gibbs-White posted a photo on social media on Monday, thanking people for their messages and showing the stitches across his forehead and nose.
Sanchez also posted a photo of some stitches at the top of his head and added in reference to Gibbs-White: "Seen that you came out worst [worse] than me, hope you are OK big man."
Pereira said that Gibbs-White's substitution was not related to concussion, so he is not subject to strict protocols surrounding such head injuries.
That means the 26-year-old's availability for Thursday is only dependent on whether the cut on his face heals in time.
Had Gibbs-White been judged to be concussed, he would probably have been ruled out. Players must not return to activity for at least 24 hours, before following a 48-hour period of rest and six-stage "return to play" process.
Premier League teams are permitted to make one concussion replacement per match, which do not count towards their usual substitution limits. Chelsea used that dispensation during Monday's match for youngster Jesse Derry.
The 18-year-old, who was making his debut, clashed heads with Forest's Zach Abbott and was taken to hospital. Chelsea said he was "conscious, talking and undergoing precautionary checks".
Brain injury charity Headway's chief executive Luke Griggs praised the care given to Derry and Abbott but added that the subsequent collision between Gibbs-White and Sanchez "shows that head injury protocols need to go much further".
"The use of concussion substitutions is certainly welcome, though with each team only receiving one permanent concussion substitute per game, clubs are limited as to what they can do when additional head injuries occur and frustration remains with football's unfathomable refusal to allow temporary concussion subs," said Griggs.
"Questions will rightly be asked about Gibbs-White and Sanchez not being subjected to the mandatory return-to-play protocols, due to them not being replaced as concussion substitutes.
"This is not about the clubs, which clearly provided first-rate treatment to their players, it's about the laws and protocols of the game - ensuring player welfare drives decision-making."
