The Uruguay international is available on a free transfer after an injury-hit two years at Old Trafford, but the veteran has been linked with a wide number of teams both in Europe and South America.
According to Marca, Villarreal are looking to win the race for the 35-year-old as they bid to earn a return to the Champions League.
Despite reaching the semi-finals in 2021-22, the Yellow Submarine will compete in the Europa Conference League this season, a consequence of a seventh-placed finish in La Liga.
However, the report claims that Villarreal cannot tie up a deal for Cavani until they have parted ways with Paco Alcacer, something which means that the completion of any transfer is not imminent.
Meanwhile, it appears likely that Georginio Wijnaldum's stint in the Paris Saint-Germain squad will end after just one season.
Although the Netherlands international made 37 appearances in all competitions last season, the 31-year-old was never regarded as a first-choice option.
Furthermore, PSG have already added Vitinha to their midfield this summer with Renato Sanches another potential signing for later in the transfer window.
According to Calciomercato, AC Milan and Roma have both been offered the opportunity to sign the player, initially on a loan deal.
The report suggests that much will depend on whether PSG are willing to make a contribution to Wijnaldum's wages during any temporary stint elsewhere.
Over in Spain, Barcelona are looking to part ways with a number of their fringe players over the coming weeks as they bid to free up space on their wage bill.
Goalkeeper Neto features among that list with the 33-year-old happy to end his three-year stint at Camp Nou in search of first-team football.
According to Sport, the Brazilian stopper has been offered to Napoli, who are in the market for a new goalkeeper after the departure of David Ospina.
The report suggests that head coach Luciano Spalletti is currently contemplating the opportunity with other options also available to the Italian side.
At one stage, Neto appeared likely to be on his way to Celta Vigo, but it is claimed that an agreement could not be reached.
On the back of scoring 21 goals in Ligue 1 last season, Rennes attacker Martin Terrier is inevitably attract interest from clubs across Europe.
Liverpool are said to be considering whether to move for the 25-year-old in the event that Roberto Firmino departs Anfield this summer or in 2023.
Nevertheless, according to L'Equipe, West Ham United and Leeds United are also monitoring the situation ahead of potentially launching bids of their own.
The report claims that Rennes would look to recoup in excess of €40m (£34.06m) if any of the Premier League clubs were to make an official move over the coming weeks.