The 83-year-old was part of Jock Stein's famous Hoops line-up which became the first British club to lift the European Cup with victory over Inter Milan in 1967.
Affectionately known by the nickname Ten-Thirty, he also won five league titles, three Scottish Cups and four League Cups over two stints at Parkhead.
The club said in a statement: "Bertie Auld's family would like all Celtic fans and football fans in general to keep Bertie in their prayers as they confirmed today that he is suffering with dementia.
"The family would like to thank everyone for their huge support and request privacy during this difficult time.
"Bertie is being well cared for at his own home, surrounded by all the family.
"Everyone at Celtic would like to add their best wishes to Bertie and his family. Bertie is a true Celtic icon, one of our greatest sons and someone the club and our supporters love and respect dearly.
"We will continue to give Bertie and his family all our love, care and our ongoing full support at this hugely challenging time."
Former Celtic striker Chris Sutton, whose father Mike died last year of dementia, was among those to react to the news on Twitter.
He posted: "Really upsetting this is. Dementia is such a cruel disease. Bertie is a true legend. Brilliant player and one of the funniest guys ever. Stay strong Bertie."
Ateltico Madrid ace Moussa Dembele, who spent two-years with the Hoops in Glasgow, added: "Support and love to a top man and Celtic legend and his family."
Auld made 279 appearances for Celtic over two spells and also starred for Dumbarton, Birmingham and Hibernian during a glittering playing career before moving on to a career in management.
He remains a hugely popular figure with the Hoops support and was a regular at Celtic Park before the pandemic struck.
Billy McNeill - the man who skippered Celtic to their historic win at Lisbon's Estadio Nacional - died in 2019 after being diagnosed with dementia.