McClaren takes after leaving Manchester United following two years on manager Erik ten Hag's staff.
It marks a return to international football for the former England coach.
McClaren has signed a two-year deal and said: "I had no hesitation in accepting a position that is ideally suited to me at this stage of my career.
"I quickly came to realise the potential the country possessed.
"In the years since, that potential has grown significantly and there is every reason to believe that with continued hard work and development a bright future lies ahead."
Ten Hag said McClaren had been "an invaluable source of support and advice".
"Steve and I have been friends for 16 years since we first worked together at FC Twente and that relationship will continue as he embarks on this next challenge, as will his bond with Manchester United," he said.
McClaren added: "I wish to say a huge thank you to everyone at Manchester United for their friendship, help and support during the past two years."