"The Premier League was a big draw for me and it's where I aspire to be after the summer I've had [at Euro 2016]," Hendrick told the club website after signing a three-year deal.
"Burnley did so well last year going unbeaten for five months and how much they wanted me this summer was a real positive for me.
"I know there have been a number of bids and I was sitting there waiting for one to be accepted.
"I would have preferred it to happen earlier to and get to know your teammates and coaching staff a bit more, but I am just thankful it's done now and I can go away with Ireland knowing my future is settled.
"It's a little weird signing for a club and not meeting everyone for more than a week, but I am really looking forward to getting back and training with the lads."
"After a protracted deal, we have got a main target of ours and we welcome Jeff to the club and hope he does well," he said.
"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."