The ex-Manchester United midfielder is said to be mulling over a £130,000-a-week contract to play under Roberto Mancini, his former team-mate, at Eastlands.
Veron, now captain of his hometown team Estudiantes in La Plata, played alongside Mancini at both Sampdoria and Lazio.
City slicker: Veron could be set to link up with Mancini again
La Plata newspaper Hoy claims that the new City boss has directly approached Veron with the intriguing offer.
It is reported that Veron initially dismissed the thought of returning to the Barclays Premier League, where he endured a disappointing six-year spell with United and later Chelsea, following a £28m move to Old Trafford in 2001.
However, he has been asked to reconsider because the fee is so high that it is would pay for Estudiantes new stadium to be completed and is the equivalent of 10 years of operating losses for the Argentine club.
The 34-year-old last week retained the award for the South American Player of the Year, marking a stunning Indian Summer for Veron.
After leading Estudiantes to glory in the Copa Libertadores in July, he was named the most valuable player in the tournament, which is South America's equivalent of the Champions League.
Veron was also a fixture in Diego Maradona's limping march to next summer's World Cup finals and is expected to be in Argentina's starting XI at South Africa 2010, despite being a critic of the boss.
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