The Australian international was a target for Ange Postecoglou's side but after opening talks with the midfielder they were gazumped by the English Championship outfit.
Postecoglou confirmed Celtic had made an approach, stating: “With Riley McGree there weren’t a lot of discussions. There was a bit of interest but when it’s clear to me that they don’t want to come to us, it’s fine. We don’t progress it from there."
The 23-year-old has instead signed a three-and-a-half year contract with Middlesbrough who have poked fun at Celtic by posting a picture of their new arrival on Twitter wearing the club's dark green away kit, along with the message: "We've gone for the away kit celebration shot this time.
"Some people prefer red to green, but hey ho..." folowed by a winking face emoji.
The seven-times capped international started his career in Australia with Adelaide United, and he also had spells with Club Brugge, Newcastle Jets and Melbourne City.
He spent the first half of this season on loan at Birmingham City from MLS side Charlotte, but he has now moved to Teesside on a permanent transfer.
Boro manager Chris Wilder said: “We’re delighted to bring Riley into the club. He’s another good young player who wants to do well, and he will fit in well with us.
“He did well during his loan at Birmingham, we’ve been looking at him for a while, and when the opportunity came up for us to bring him to the club we jumped at it.”
Head of football Kieran Scott added: “Riley is a player who fits the mould of what we are looking for in that he is young, hungry, full of potential, and has a desire to join Middlesbrough.
“He turned down other opportunities because he wanted to join us here and be a part of what we are striving to build.”
McGree posted his own Twitter message, writing: “Huge honour to sign for @Boro, cant wait to get started! #UTB.”