The 29-year-old, who has featured 47 times for the Magpies since arriving from Aston Villa in 2022, makes the switch to Teesside in search of regular football after overcoming injury setbacks in recent years.
He joins a Boro side who have started the season strongly under Rob Edwards and currently occupy second place in the table after winning all three of their opening Championship fixtures.
Everybody at Newcastle United wishes Matt well during his time at Middlesbrough.