The club are seeking clarification over who makes the decision after being told different things by local and Scottish governments.
A statement read: "Celtic Football Club is extremely disappointed and concerned that approval could not be given at today's meeting of the Safety Advisory Group (SAG) to grant permission to install rail seating at Celtic Park, designed to allow safe standing by supporters.
"The club has provided substantial and detailed information in support of this proposal over the past two-and-a-half years and has made every effort to address comprehensively all issues raised by the SAG.
"Today's decision has been made despite the club also providing the requested additional information from an independent stadium safety expert, Dr Steve Frosdick.
"Surprisingly, we have also today been informed by the Safety Advisory Group, chaired by Glasgow City Council, that this may be a matter for the Scottish Government, and yet we have previously been told by the Scottish Government that this is a decision which must be taken locally at council level.
"Spectator-safety at Celtic Park is of paramount importance and we believe the proposals we have made would significantly enhance safety at matches. This is a vitally important spectator-safety issue. It is also an opportunity for Glasgow and for Scottish football to lead the UK in this area.
"We do not understand the rationale for today's decision and in an attempt to seek clarity on this matter, we have requested a full written explanation for the judgement made. Once this has been received we will be in a position to communicate further with our supporters."
Top-flight clubs in Scotland relaxed their rules on safe standing in December 2011 but none have introduced official areas.
Supporters Direct Scotland stated its disappointment with the decision.
A statement read: "Research conducted recently by SDS in association with the SFA showed strong support for safe standing in Scottish football.
"Over 90 per cent of respondents stated that they would be in favour for the reintroduction of safe standing areas, whilst the majority of those surveyed said that they would be more likely to attend games on a regular basis if safe standing was available."