Manchester United, Liverpool, Chelsea and Arsenal have all held initial talks with Paris St-Germain for Germany international Draxler, according to The Gambling Times.
The 23-year-old faces being pushed down the pecking order at PSG following the world-record signing of Neymar from Barcelona.
But Draxler’s agent Roger Witterman has attempted to play down speculation over his client’s future.
Witterman said: “Draxler will stay and will not be sold this summer.
“Paris is building a group with the best players in the world at every post, and Julian is one of them and he wants to continue to be part of it.”
How Liverpool could line up with Julian Draxler
Draxler has been linked with a move to Arsenal ever since bursting onto the scene at Schalke as a teenager.
Arsene Wenger is on the hunt for a winger but is struggling to land top target Thomas Lemar from Monaco.
Draxler could be a more realistic target than Lemar or Leicester’s Riyad Mahrez, with time running out to complete signings.
But it is Liverpool, scouring the market for Philippe Coutinho replacements, who are the favourites to sign Draxler.
BetStars Director of Trading Ian Marmion explained: “Liverpool have emerged as the clear 11/8 favourites to sign PSG’s Julian Draxler, after being told he is no longer needed at the Parc des Princes.
“The World Cup winner only joined the Paris side in January but since the arrival of Brazilian star Neymar, Draxler has become surplus to requirements.
“The German international will be the creative influence that Liverpool need should they sell Coutinho to Barcelona.”
United boss Jose Mourinho, like Wenger, is on the hunt for a winger.
The Red Devils have been unable to sign Ivan Perisic from Inter Milan and it has emerged that they are now considering Lemar.
Chelsea head coach Antonio Conte is also desperate to bolster his ranks after a miserable summer in the transfer market so far.