Spurs, West Ham, Juventus and Roma are showing the strongest interest in the Belgian forward, who has scored 19 goals in 43 appearances this term.
If Marseille agree to sell their star man, Batshuayi will prioritise clubs playing in the Champions League.
West Ham can offer the prospect of home games at the Olympic Stadium next season but the Hammers are struggling to qualify for Europe's premier competition, given they sit eight points adrift of the top four.
If Tottenham make a concrete offer, their primary competition is expected to come from Juventus and Roma, with the Italian pair both currently occupying Champions League spots in Serie A.
Batshuayi moved to Marseille from Standard Liege in August 2014 and the 22-year-old feels a strong sense of gratitude to the French club for giving him his chance to shine in a top European league.
The youngster will not force a transfer this summer but Marseille are 16th in Ligue 1, just five points above the relegation places, and are understood to be open to cashing in on their prize asset.
According to reports in France, Marseille want the fee to exceed the £24million they sold Didier Drogba to Chelsea for in 2004.
Tottenham have relied heavily on Harry Kane this season and are keen to bring in attacking cover
Kane has started all of Spurs' 34 Barclays Premier League matches, scoring 24 goals in the process
Source: PA