The 34-year-old Estudiantes captain told the radio station La Red that he was phoned by Mancini, a former team mate of his in Italy and his coach at Inter Milan, to make the offer.
Mancini took charge of Manchester City last month.
"I presented the offer to the directors of the club (Estudiantes) so they could consider it, maybe it appeals to them. For my part there isn't much thinking to do, my (negative) answer was instantaneous," Veron said.
"It's not possible at the moment and it's not a question of money ... My family and my affections are involved," he said.
Veron has a strong bond with Estudiantes, whom he led to the South American Libertadores Cup title in July and second place behind Barcelona in the Club World Cup in Abu Dhabi last month.
His performances for Estudiantes and Argentina have seen him voted South America's top player for the last two years after a glittering career in Europe, especially Italy, though he also played for City's local rivals Manchester United.
Veron said he returned to his home town of La Plata in 2006 to see out his career with Estudiantes and try to lead them to more trophies after his father Juan Ramon helped the team lift the Libertadores Cup three times from 1968 to 1970.
"I took a decision and I want to respect it. I can't leave, I'd be going against my convictions, I'd be letting myself and the fans down above all else," said Veron, who donates part of his salary to Estudiantes' youth scheme.
