Targett has clocked up over 200 career appearances, the majority coming in the top-flight and has featured 47 times for the Magpies since arriving from Aston Villa in 2022.
He is the seventh signing of the summer, following Alfie Jones, Callum Brittain, Abdoulaye Kante, Sontje Hansen, Sverre Nypan and Adilson Malanda to the Boro who are currently 2nd in the Championship after winning all three of their opening games.
The left-back made his Champions League debut in the 2023/24 season and featured under Newcastle manager Eddie Howe in preseason, scoring in a friendly against Espanyol.
The left-back was facing a season as a back-up player at St James' Park and it is clear that after talks with Howe, he decided it was time to leave the side in what is effectively a goodbye to the veteran defender.
A Newcastle statement read: "Everybody at Newcastle United wishes Matt well during his time at Middlesbrough.”