The former Leicester City player worked with Celtic's manager Brendan Rodgers while they were together at Reading.
"It was good to see Brendan," Hayes told the Celtic website (www.celticfc.net) after agreeing a three-year contract.
"I still remember how well he looked after me when I went over to England. He’s a great man and a great manager to be working for."
British media reports put the fee at just over one million pounds.
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Celtic won the Cup final 2-1 to complete a domestic treble in Scotland.
Their midfielder Ryan Christie will join Aberdeen on a season's loan as part of the deal.
(Reporting by Steve Tongue, editing by Pritha Sarkar)
