The People claims that City will offer Barcelona £50m for the World Cup winner, with wages of £8m-a-year over five seasons.
Pique's agent is reportedly aware of the interest, while the Primera Division giants are apparently bracing themselves for an offer for their centre-back.
Pique started his career at Barça, before moving to City's rivals Manchester United in 2004. He returned to Camp Nou in 2008 and has since become an important figure for club and country.
Mancini already has a wealth of options at his disposal in central defence, including Kolo Toure, Vincent Kompany, Micah Richards and Joleon Lescott.
The report comes off the back of the announcement that City recorded losses last year of £121m, with chief executive Garry Cook indicating there would be a curve in spending on the transfer market in the future.