The pair appeared to clash on the pitch during last week's 2-0 victory over Lyon.
In the second half, the Ligue 1 giants were awarded a penalty and both players wanted to take the spot kick, but it was Cavani who eventually took the shot and missed.
According to El Pais, chairman Nasser Al-Khelaifi tried to sort out the dispute between the pair by offering Cavani €1m (£877,257) to relinquish his role as regular penalty taker and give it to Neymar, who joined the French club from Barcelona for a world record £200m fee in the summer.
The bonus is believed to be a clause in Cavani's contract which gives him a financial boost if he finishes the season as Ligue 1's top goalscorer, but the Uruguayan declined the offer.
However, according to Le Parisien, PSG have rejected these reports, insisting that their forward has not been asked to give up penalty duties in exchange for money.
Cavani and Neymar have scored 10 league goals between them this season.