The 39-year-old, who joined Villa in 2021 after being assistant to Vincent Kompany at Anderlecht, took charge of two first-team games October after Steven Gerrard’s sacking.
The coach has previously managed England and Albion Under-21s and took to social media to bid his farewell to Villa.
He said: “My time at Aston Villa has come to an end.
“I have loved the experience that the club gave me and the many people I have had the privilege to work with and get to know personally.
“To the fans, it was an honour and privilege to represent you on the touchline,” he added.
Danks joins Jonathan Woodgate and Grant Leadbitter under recently-appointed Boro boss Michael Carrick.