Marco Silva's side are looking to bring in a replacement for Joao Palhinha, who recently competed a big-money move to Bayern Munich, and Andre is believed to be the London club's leading target.
According to the London Evening Standard, Fulham have stepped up their interest in Andre, who is thought to be valued by his Brazilian club in the region of £25m.
The 23-year-old, who was also linked with a move to Craven Cottage last year, is a five-time Brazil international.
The midfielder has represented Fluminense on 187 occasions in all competitions, scoring four goals and registering four assists in the process.
Fulham have also been linked with Manchester United's Scott McTominay, while the club are said to be admirers of Arsenal attacker Emile Smith Rowe.
Everton 'in talks' with Napoli over Lindstrom
Elsewhere, Everton are said to be in discussions with Napoli over a deal for attacker Jesper Lindstrom during this summer's transfer window.
Transfer expert Fabrizio Romano claims that the Toffees want to take Lindstrom on loan for the 2024-25 campaign, with a buy clause included in the agreement.
There is allegedly "still work to do" when it comes to a possible deal, but Napoli are open to letting him move to Goodison Park this summer.
The 24-year-old made the switch to his Italian club from Eintracht Frankfurt last year, but a disappointing 2023-24 campaign saw him make 29 appearances without scoring or providing an assist.
However, Lindstom managed 14 goals and 14 assists in 80 appearances for Frankfurt, while he contributed 15 goals and 15 assists in 65 outings for Brondby earlier in his career.
Southampton 'fail' with loan offer for Rak-Sakyi
Meanwhile, according to the London Evening Standard, Southampton have failed with a loan bid for Crystal Palace attacker Jesurun Rak-Sakyi.
The 21-year-old's future is currently the subject of much speculation, with a number of clubs in the Championship thought to be exploring the possibility of signing him.
Southampton allegedly recently made an offer to loan the forward for the 2024-25 campaign, but Palace were quick to reject the bid from their Premier League rivals.
Rak-Sakyi came through the youth system at Chelsea before heading to Palace in 2021, and he has made 12 first-team appearances for the Eagles, including six Premier League appearances last term.
The attacker also had an impressive loan spell at Charlton Athletic during the 2022-23 campaign, finding the back of the net on 15 occasions in the third tier of English football.