The Chile international has emerged as one of the hottest properties in European football with a host of clubs, including City, Barcelona and Manchester United, all battling for his signature this summer.
City are understood to have tabled the highest bid for the 22-year-old - reported to be in the region of ?30million - but Udinese value their prize asset at a staggering ?44million.
Barcelona have rivalled the Eastlands outfit's interest in Sanchez and now Mancini has called time on his pursuit of the gifted wide man, despite holding positive initial talks with the player.
Mancini told Sky Italia: "I talked to Sanchez, he said he was willing to come to City.
"But after the last offer we are out of the race for the Chilean player."
Udinese, meanwhile, confirmed the 22-year-old winger would prefer a move to Spain rather than England.
"Have City pulled out of the race?," Gino Pozzo, son of Udinese owner Giampaolo Pozzo, said to Corriere dello Sport. "You should ask them. We have not entered negotiations with City.
"The will of the player is to go to Barcelona. The negotiations with the Catalan club are going forward.
"With calmness, we will find a solution. We will speak to Barca and we will see."
Pozzo has not ruled out the possibility that Sanchez, who is valued at 50million euros (?44.95million), could remain at Udinese for a further season.
"He (Sanchez) said it yesterday and we have reiterated it in the past," Pozzo said."It is an option."
Meanwhile, Mancini is confident Carlos Tevez will remain at the club next season despite the Argentina forward recently expressing his desire to leave the club.
Tevez, who played a pivotal role in the Eastlands club winning the FA Cup and securing a top-four Premier League finish, has been strongly linked with a move to Italy this summer.
"Tevez is a good player," Mancini added. "I believe he will remain with us."
The 27-year-old is under contract with City until June 2014 and is a reported target of Inter Milan.
Source: Team Talk
Source: Team Talk