Davies was assistant under Viera and will take up his new role when the Frenchman starts work as head coach of New York City next month.
The 41-year-old, who had an unsuccessful eighth-month stint as Chester boss in 2008, has rebuilt his career over the past four years.
He started working with the Premier League club in 2010 and progressed to become Veira's number two in June 2013.
"It was an honour to work with Patrick and it’s a privilege to take on the task of continuing the fantastic work he’s undertaken with a talented and dedicated group of players," Davies told Manchester City's website.
"I’d like to take this opportunity to thank Patrick for his mentorship and guidance over the past two and a half years and to thank Txiki Begiristain and everyone at the club for this fantastic opportunity.
"I’ve learned a great deal as Patrick’s Assistant and I’m delighted to be taking this next step in my coaching career to play a part in helping to shape the footballing future of this great club.
"Our first goal is to produce players for our first-team and we have had a great start to the season in this respect with many of the EDS squad training and playing competitive football with the seniors on a regular basis.
"I look forward to this continuing over the coming weeks, months and years."
Winsford-born Davies, who won one cap for Wales, started his career at Manchester United, making a handful of first-team appearances. He had spells with Luton Town, Macclesfield Town and Rochdale before moving into Welsh football with Bangor City, TNS and Rhyl.
He arrived at Chester in summer 2006 and coached the youth team. He had one game as caretaker manager in April 2007 following Mark Wright's departure.
Davies continued to work with the youth side until March 2008 when Bobby Williamson was sacked as manager and Stephen Vaughan put him in charge until the end of the season.
He managed to save the club from relegation and was given the role on a permanent basis, signing a two-year contract.
However, he was sacked in November 2008 following a disappointing start to the season with Wright replacing him.
Davies returned to the club the following month when he was appointed head of youth and worked as assistant manager under Mick Wadsworth, leaving when he departed in September 2009.