The 22-year-old has been linked with Arsenal, Chelsea, Tottenham and Newcastle (by The Sun), but it is the Eagles that are hoping to swoop in ahead of their Premier League rivals.
Like table-toppers Leicester City, Alan Pardew's men are upsetting the apple cart this season, and thus, may be looking to an ambitious January move as an opportunity to cement their place in race for a European spot.
Palace's draw with Everton on Monday night sent the club sixth, but with the six sides below them all within two points, Pardew will know that their position is precarious if they are not able to maintain their form.
Their biggest problem has been finding a striker that is able to put the ball in the back of the net. Despite having several forwards in their squad - with four included on the bench against the Toffees - a natural striker has not scored for the side in the top flight this season.
And while Connor Wickham's hold up play has been exemplary in recent weeks, the club really do need a goalscorer if they wish to stay in the top six.
Batshuayi would undoubtedly be the kind of signing that could elevate Palace to the next level. He has scored 10 times in 17 Ligue 1 games for Marseille this term, and has bagged goals in both of his appearances for Belgium.
However, before Palace fans begin to consider the possibility of Batshuayi arriving at Selhurst Park, Talksport reported last month that the French side's president Vincent Labrune had insisted that it would cost around £35m if a club wanted to take him away from Stade Velodrome, which would mean that Palace need to up their offer considerably if they want their man.