The 23-year-old is in the final six months of his contract with the Spanish top-flight side and has been a regular in La Liga so far this season.
However, The Sun on Sunday report that a £1.5million deal with the Canaries could be struck during the current transfer window, rather than risk losing him for nothing in the summer.
Quintilla came through the youth system at Barcelona and earned youth caps for Spain through to under-19 level, with the article claiming he is keen on a Premier League move.
It is also stated that City's precarious position in the survival race will be factored into his contract - claiming that the Canaries could also be open to offers for Northern Ireland left-back Jamal Lewis.
The Spaniard has started 15 of Villarreal's 19 league matches so far this season, including a 2-1 win at Real Sociedad last week, claiming two assists.
Sam Byram has started nine of the last 10 Premier League matches at left-back, keeping the position since Lewis picked up a minor knee injury.
The £750,000 summer signing is primarily a right-back however and there has been a surge in transfer activity surrounding talented right-back Max Aarons this month, with Sky Sports recently adding Borussia Dortmund to the lengthy list of big clubs apparently interested in the England U21 international.
Sporting director Stuart Webber has stated that any permanent signings this month would have to make financial sense for City, with the threat of relegation looming large.
At £1.5m and 23 years old Quintilla could fit into that category, with a chance to impress in the Premier League potentially offering the chance for a quick profit in the summer, if City were to go down.