The First Minister answered back to the reigning Scottish Premiership champions, who released a short statement on Monday saying they had received permission from the Scottish Goverment and Joint Response Group to fly out to the Middle East. They were then backed up by the league over their version of events.
Sturgeon, however, insists the advice given to the governing bodies regarding travelling to training camps was two months ago, with the impact of the coronavirus pandemic in Scotland significantly different to what it is now.
She questioned whether travelling was the “appropriate” action to take for the Parkhead club in the light of tier 4 Scotland areas, including Glasgow, going into a second full lockdown.
She said: "As I understand it, we gave advice to the SFA about the rules around elite training camps back in November last year.
"The world and the pandemic has changed quite a bit since last November, it's not our role to sign off, to agree, to not agree in terms of what a football club does in terms of training camps.
"We set out what the rules are and the rules have been, and they may have to change due to the nature of the pandemic, that elite sports can go overseas for training camps if that is important in context of their training for competitions.
"That's, as I understand it, what we set out as the rules.
"For me, the question for Celtic, and I will try and be diplomatic here, is not so much are they overseas at training camp, it may be given the changed circumstances, that won't be appropriate. That's not really the question.
"What is the purpose of them being there? I've seen a comment from the club that said it's more for R and R than it is for training.”
The SNP leader also questioned whether Celtic players and staff were complying with social-distancing measures on their trip away after leaked photos purported to show manager Neil Lennon and captain Scott Brown socialising by the pool.
She continued: "I've also seen some photographs, and I can only comment on what I've seen as I don't know the full circumstances, that would raise a question in my mind of whether all the reams of what elite players have to do within their bubbles, if social distancing is being complied with.
"There are things that should be looked into.
"Elite sport and this is not just about Celtic of football, elite sport is in an elite position at many points over the past year doing things that many of the general public can't do.
"As long as that is the case, as we all want our sport teams to do well and train properly, as long as that is the case it's really important that don't abuse it and use it for the purposes that was intended.
"And that applies to Celtic as it would to anyone else in that position."