Arsene Wenger is reportedly willing to spend some money in January with several of his squad now injured.
The 18-year-old is being monitored by several Premier League sides - including Chelsea and Liverpool - as well as Spanish giants Real Madrid, The Sun claim.
However, with the Gunners' injury list seemingly extending by the game - with Arsene Wenger confirming that Alexis Sanchez, Laurent Koscielny and Santi Cazorla all collected knocks in yesterday's draw with Norwich City (via BBC Sport) - it is the north London side that have apparently placed themselves at the front of the queue for Neves' signature.
The Sun add that Arsenal plan to offer £12m up front to the Portuguese outfit, with a further £10m coming in add-ons.
With Francis Coquelin out for three months (via BBC Sport), it is imperative that Arsenal add a new holding midfielder if they wish to build on their solid start to the season.
The Frenchman limped off during the early stages of their game against West Brom, which left the back four exposed, and was one of the reasons why the side left the Hawthorns empty-handed.
Similarly, Coquelin's absence against the Canaries was obvious as the Gunners conceded a fairly soft goal to cancel out Mesut Ozil's opener.
While Neves would provide the steel that Coquelin has brought to the Arsenal team since his breakthrough last season, he also embodies the Gunners' philosophy of using the ball creatively. His passing is becoming one of his greatest strengths, so he would be able to take some of the pressure away from the members of the back four who lack composure on the ball.