Videos have circulated on social media apparently showing individual supporters, in different incidents, pushing locals into a fountain at Placa Reial, just off La Rambla.
But another video of undercover police arresting fans appears to show one officer wearing an old Celtic tracksuit top.
The black top has a fluorescent yellow Nike logo on the right chest, and the Celtic club crest on the left, and looks like training wear from the early 2010s era.
Police in the city forced all fans back on to the main tourist strip before walking them up the street to Placa de Catalunya, a main transport hub, where they dispersed following the incidents.
Merseyside Police are understood to be aware of the existence of the videos, as, apparently, is Moore.
"We proudly sing that we've conquered all of Europe. But let's treat this beautiful city with the respect that it deserves, and act in a manner that is befitting of LFC," the Reds chief executive wrote on Twitter.
"By all means have a good time, but we are Liverpool, and as such, let's visit here with grace and humility."
On Tuesday, as thousands of fans began to arrive in the city for the Champions League semi-final first leg, Merseyside Police's dedicated Twitter account for LFC fans tweeted the advice: "Eat, drink and enjoy all areas in the city centre. Respect monuments."
Liverpool, as a club, condemned the behaviour as "totally unacceptable".
"Liverpool Football Club is working with Merseyside Police and the authorities in Spain, who are endeavouring to identify those involved in the incident," said a statement.
"Such behaviour is clearly totally unacceptable and will not be tolerated.
"It would be inappropriate to comment further while the situation is ongoing other than to confirm the club will follow due process in any and all cases of this nature."
Merseyside Police confirmed six people had been arrested for public order offences in an unrelated incident.
"We are liaising with Spanish Police, the British Embassy, UEFA, the Football Intelligence Unit in Merseyside and Liverpool Football Club, who are all working to keep fans safe and ensure an enjoyable experience in Barcelona," said Superintendent Paul White.
"We want to reassure the public that we can act on incidents using British Law and we will take action. This type of behaviour is wholly unacceptable and I want to remind fans who are travelling with tickets to monitor official fan advice on the club's official website and social media feeds for information to assist their visit."