The Premier League duo have been tracking the 21-year-old from some time, and were believed to be readying summer bids for his signature.
City reportedly spoke to Neymar after England’s recent friendly against Brazil, whilst Chelsea are rumoured to be highly interested in the attacker as they look to usher a new generation of top talent into Stamford Bridge.
But, they both appear to be set to miss out on the Santos star, with Bayern readying a ginormous £86m offer.
The Bundesliga giants are said to have vast sums available to spend this summer as they look to support Pep Guardiola, who will take control of the club later this year, in the transfer market.
Although both City and Chelsea are heavily backed, they are unlikely to be able to match the Bavarian side.
It has been widely speculated that Neymar would remain in South America until after the 2014 World Cup, but such a large offer would tempt both the youngster and his employers to alter their stance.
Bayern are also said to be interested in Liverpool’s Luis Suarez, but any move for the Brazilian international would, likely, end their pursuit.