Gomes, 25, is expected to leave Molineux after Wolves' relegation to the Championship was confirmed in April. He has been one of the few consistent performers in a difficult season, making 38 appearances across all competitions and contributing one goal and three assists.
Wolves are asking for offers in the region of £40 million, with the Brazilian under contract until 2030.
BBC Sport reports Crystal Palace held early contact with Gomes' agency this week. The move is at an exploratory stage, but the timing is deliberate — Palace are aware that acting before the window opens on 15 June could prove decisive in a race that currently has Atletico Madrid as strong favourites.
Atletico opened talks with Wolves over a deal earlier this month but are simultaneously tracking Ederson at Atalanta — and that dual interest may be what gives Palace their opportunity. If Atletico commit to Ederson, Gomes becomes available for clubs willing to move quickly and meet Wolves' valuation.
Palace have their own pressing midfield need. Daichi Kamada, who scored in their 3-1 first-leg Conference League semi-final win over Shakhtar Donetsk, is leaving when his contract expires this summer. His departure leaves Oliver Glasner's side short of a dynamic, combative option in the middle.
Gomes fits that profile precisely. His pressing intensity, passing range, and box-to-box output represent exactly the profile Palace have lacked, and he has already proven he can perform at Premier League level across three seasons.
Manchester United and Liverpool have also been linked with Gomes previously. With the window not opening for six weeks, the picture is likely to shift further before any club makes a formal bid.