South African Gary Kirsten has been tipped as a potential successor, but Harmison believes that Giles, who is currently in charge of both England one-day teams, would be the perfect fit.
"I'd like to see them give it to Ashley Giles," he told reporters. "I think he deserves a chance. I don't see too many legitimate candidates putting their hands up and saying 'I deserve a chance to coach England'.
"There's talk about Gary Kirsten. I don't think Gary Kirsten will want the England job because of the way he left India and South Africa: he wanted to spend more time with his family, and he's not going to get that in England.
"I think Ashley Giles deserves a chance to put his stamp on this side: I'd give him the Test job and the one-day job and possibly detach the Twenty20 job altogether. He's got all the trades. He worked hard as a cricketer, and as a coach he did all the hard work at Warwickshire, winning the Championship. He'll throw everything behind it, and I think he'll succeed - especially with people like Alastair Cook around him."
Giles took 143 Test wickets at an average of 40.60 during his Test career with England.