The incident happened in Janefield Street at Parkhead before Sunday’s match against Rangers as fans tried to get into the stadium before kick-off.
Police Scotland said one person was taken to hospital after falling from a wall, while four others were treated by first-aiders at the scene.
A spokesman for the club said: “The safety of our fans will always be of paramount importance and this is a matter which we are treating with the utmost seriousness.
“The club, along with Police Scotland, will be investigating this matter fully to identify the causes and take the appropriate steps to ensure this does not happen again at Celtic Park.
“The club and Police Scotland will advise of the outcome of this investigation once it is complete.
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“We are fully aware of the serious difficulties which arose yesterday for supporters, and we sincerely regret and apologise that so many fans were faced with this situation.”
He added the match was the first time new segregation and access arrangements had been used.
Victoria Martin has held a season ticket with the club for more than 20 years and said fans were being left to “fend for themselves”.
The 33-year-old stylist from Glasgow said: “I have been to games all over Europe, you really don’t expect to be caught up in this situation.
“Something really, really bad could have happened.
“There was no guidance - the police were of no help whatsoever.”
Chief Superintendent Brian McInulty said on Sunday: “This was a dynamic situation, occurring 10 minutes before the match started.
“Officers and stewards reacted quickly upon realising there was an issue and put in place measures to relieve the congestion.
“This included putting in place cordons to prevent further entry at Janefield St and opening up London Road to allow fans to access the stadium from the south.
“We work closely with Celtic Football Club to ensure the safety of all fans attending matches.
“We plan and practice various scenarios to ensure that if an incident occurs it will be dealt with as quickly as possible, as happened in this case.
“We will review today’s incident and work with Celtic to ensure any learning is quickly put in place.”