Neves came through the ranks at Benfica and has played 75 matches since breaking into the senior team for his boyhood club, scoring four goals and providing three assists in that time.
Last week, Benfica president Rui Costa revealed that the club had received an official bid for the 19-year-old rising star, who has earned comparisons to Euro 2024 winner Rodri.
Costa also admitted that Neves could be sold for less than the £101m release clause built into his contract this summer.
PSG closing in on Neves' signature
While the Benfica president did not mention who the interested party was, widespread reports suggested the PSG had made signing Neves their main priority this summer.
The Ligue 1 champions were also believed to have enquired about signing the 19-year-old earlier in the transfer window but were unwilling to meet his release clause.
However, according to transfer guru Fabrizio Romano, the clubs have had a round of positive discussions today, and the French champions are now aiming to finalise a deal for the midfielder next week.
The report adds that Les Parisiens are discussing an offer which would see them pay around £60m in cash, while also including Renato Sanches in the deal.
It is understood that the PSG have already made contact with Neves, who has confirmed his interest in joining the club, and has agreed on personal terms.
The Portugal international's current contract at the Estadio da Luz runs until the completion of the 2027-28 league season.
Neves to PSG spells good news for Man United's Ugarte hopes
Man United have been pushing ahead with their transfer plans this summer since the completion of the Euros, with Joshua Zirkzee and Leny Yoro arriving at Old Trafford in the last few weeks.
The Red Devils were reportedly considering a move for Neves earlier this summer but have since turned their focus to PSG midfielder Manuel Ugarte.
Thus, PSG's reported progress in signing Neves is likely to excite the United board, as it may suggest that Les Parisiens will be open to allowing Ugarte to leave this summer.
Ugarte's agent Jorge Mendes, who also represents Yoro, is believed to be pushing for his client to secure a move to Old Trafford in the near future.