The Premier League new boys have not disclosed the fee for the Republic of Ireland but it is understood to be in excess of £10m.
Hendrick, who impressed for the Republic of Ireland at Euro 2016, has penned a five-year contract with the Clarets.
“After a protracted deal, we have got a main target of ours and we welcome Jeff to the club and hope he does well," said Burnley boss Sean Dyche.
“He is someone who can add to our strength in midfield and he’s a player who is still learning and still developing and is someone who we think we can work for us now and in the future.