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.
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.
"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.
