The Frenchman has entered the final 12 months of his contract with Lyon, and a number of teams, including Arsenal and Manchester United, are thought to have considered moving for him in the current market.
However, according to The Sun, the strongest interest is currently arriving from Southampton, with the Saints eyeing a £10m deal for the centre-forward.
Dembele previously represented Fulham and Celtic before making the move to Lyon, while he spent the second half of the 2020-21 season on loan at Atletico Madrid.
Last term, the 25-year-old scored 22 goals and registered five assists in 36 appearances at club level, including 21 goals and four assists in 30 matches in France's top flight.
Meanwhile, according to The Sun, Fulham are considering a move for Coventry City midfielder Gustavo Hamer during this summer's transfer window.
The report claims that Cottagers boss Marco Silva is an admirer of the 25-year-old, who is believed to have been made aware of interest from the Premier League.
Hamer - likely to cost in excess of £10m - was a key player for Coventry during the 2021-22 campaign, scoring three times and registering 10 assists in 39 Championship appearances.
The Brazilian-born midfielder only signed a new deal with his current side in March, with his existing terms due to run until the end of the 2023-24 campaign.
Elsewhere, Millwall have reportedly agreed a deal with Leeds United to loan midfielder Jamie Shackleton for the entirety of the 2022-23 campaign.
The 22-year-old's future has been the subject of much speculation in recent weeks, with a number of Championship sides said to have expressed an interest in his services.
According to Football Insider, Millwall have now agreed to sign Shackleton on loan for the 2022-23 season, with the midfielder set to follow Charlie Cresswell from Elland Road to the Den.
The Englishman found it difficult to secure action under new Whites boss Jesse Marsch, but he did play on 16 occasions in all competitions last season, including 14 times in the Premier League.
Meanwhile, according to the Daily Mail, Swansea City have told interested clubs that Joel Piroe will not be allowed to leave the Welsh club this summer for anything under £20m.
The 22-year-old moved to Swansea from PSV Eindhoven last summer and enjoyed a excellent first season at the Liberty Stadium, scoring 23 goals and registering six assists in 46 appearances.
The report claims that a number of Premier League sides are showing an interest in the former Netherlands Under-20s international, but his Welsh club have set his valuation at £20m.
Leicester City are believed to be one of the top-flight clubs interested in his services, while Watford are also thought to have been in touch to discuss the Dutchman.