The 28-year-old Dubliner joins the Magpies following the expiration of his contract with Burnley, where he made 139 appearances in four seasons across all competitions and helped the Clarets qualify for the Europa League in 2017/18.
He had joined Burnley for a reported fee of more than £10m in August 2016 having made more than 200 appearances for his first club, Derby County.
Hendrick - who has won 54 caps for his country and represented them at Euro 2016 - told NUFC TV: "It's a massive club and playing over the years, going to the stadium, it's one of the best stadiums to play in, so I just want to get playing there.
"I've had over 100 games in the Premier League but I want to keep going and keep improving with a different team. It's going to be different for me, the demands with the bigger stadium, the crowd, how big the club is, so I've got to take that on and keep pushing myself and striving to get better.
"I'm looking forward to it - I can't wait."
Head coach Steve Bruce added: "Jeff is someone I've admired for a long time. I watched him when he was at Derby, then he went to Burnley for a lot of money and couldn't afford him, so I missed out then!
"He's an experienced international football who can play all across the midfield so we welcome him aboard. He'll be a big asset to us, on and off the pitch.
"He's a great pro, down to earth, and at the age of 28 it's a wonderful time to get somebody. It is the peak of his career - he's a really fit boy who has played 122 times in the Premier League for Burnley over the last four years, and I'm sure he'll be a great addition to the squad."