The 21-year-old Brazilian is widely regarded as one of the hottest properties in world football at the moment, and is expected to make a big-money move to Europe after the World Cup in 2014.
Barcelona remain favourites to secure Neymar's signature, however Manchester City's top two executives Ferran Soriano and Txiki Begiristain - former employees at Camp Nou - are said to have held initial talks with the player's family and representatives.
The duo approached the Neymar party in a Wembley Stadium box earlier this month, on the night of England's 2-1 friendly victory over Brazil, to celebrate the 150-year anniversary of the Football Association.
Etihad Stadium chief executive Soriano, and director of football Begiristain, were guests of the FA in the VIP Wembley suite on February 6, but took time out to meet those close to the South American superstar.
Plenty of hype surrounded Neymar's first appearance in this country, but despite playing 90 minutes the forward failed to impress in a lacklustre performance by the boys from Brazil.
However, that is unlikely to deter Roberto Mancini, who is keen to add him to his Manchester City squad in the future.