Manchester United saw an opening bid of £46m (€50m) turned down by the Eredivisie side.
Ajax are desperate not to lose all of their star players who helped them reach the semi-finals of the Champions League last season.
Defender Matthijis de Ligt has joined Juventus while Barcelona snapped up midfielder Frenkie de Jong.
United have been quoted £64m (€70m) if they want to snap up the Brazilian winger.
That is according to Yahoo Sport, who claim Ajax are also attempting to convince Neres to sign a new contract.
One beneficiary of any deal for Neres will be the 22-year-old’s former side Sao Paulo.
Yahoo Sport say the Brazilian outfit have a 23 per cent sell-on clause which could bring them some much-needed investment.
However, United are negotiating with several clubs in a bid to bolster their ranks ahead of Thursday’s 5pm transfer deadline.
Leicester defender Harry Maguire is in the north west today to complete his medical ahead of an £85m switch.
Talks are also ongoing with Juventus over two separate deals.
The Red Devils are attempting to buy experienced striker Mario Mandzukic.
They are also discussing the possibility of swapping Romelu Lukaku for Paulo Dybala.
However, the Argentine attacker is yet to decide if he wishes to leave Serie A for the Premier League.
Manager Ole Gunnar Solskjaer has already splashed the cash on Aaron Wan-Bissaka and Dan James this summer.
But there are more new faces expected through the door within the next five days.