While the Toffees have made steady progress up the Premier League standings since the arrival of Carlo Ancelotti, they continue to struggle for clean sheets at the back.
After losing 3-2 to Arsenal on Sunday afternoon, Ancelotti acknowledged that his team have performed poorly in defence, suggesting that he may want to strengthen in that area before next season.
According to The Mirror, Gabriel is high up on Ancelotti's shortlist after consistently impressing for Lille during the current campaign.
Having recently penned a new contract, the Brazilian could cost in the region of £30m, a fee which may put off some of his suitors.
However, the report claims that Everton are willing to meet that asking price in a bid to edge out Arsenal and Leicester City.