According to The People, the Citizens are planning an audacious £50m bid for the Brazilian striker, who has been linked with a switch to Barcelona.
Mancini remained coy in stating whether City would attempt to lure Neymar to the Etihad Stadium, but said that he cannot wait to get start working with Begiristein.
The former Barcelona director of football was unveiled in his new role last week.
"I don't know if it is true that Barcelona have already made a down-payment for Neymar," the Italian is quoted as saying. "I think when Txiki starts to work here next week we can start to talk about the future for us because this is very important.
"Neymar is a Santos player and I don't know what can happen with him in the future, but I am very happy to work now with the new sporting director because he gives us the chance to move very quickly regarding some players.
"If we work well together then the next three or four years for Manchester City can be very strong."
Neymar recently stated that playing alongside Lionel Messi would be a "dream".