Turkey international Mor only joined Dortmund from FC Nordsjaelland last summer but could leave Germany after just 12 months.
The 20-year-old made 12 Bundesliga appearances under ex-Dortmund boss Thomas Tuchel last season.
Mor's playing time at Dortmund could be reduced further next term following new manager Peter Bosz's swoops for Mahmoud Dahoud and Maximilian Philipp.
And Mor is now interested in starting afresh elsewhere.
The Sun claim Merseyside giants Everton and Liverpool are battling it out with Arsenal, Fiorentina, Roma and Inter Milan for Mor.
It is said Dortmund are willing to sell him on a permanent basis and are already resigned to losing him.
While Serie A side Fiorentina have tabled a £10.7m bid for Mor, The Sun claim he is keen to move to the Premier League.
Liverpool boss Jurgen Klopp tried to sign him last summer before Mor's move to the Westfalenstadion.
Klopp remains a fan and is now challenging Everton and the Italian giants to the youngster's signature.
The Sun add that there is also interest from Arsenal and Arsene Wenger has requested to be kept informed of Mor's situation.
Mor, who was born and spent his youth career in Denmark, has opted to represent Turkey at international level.
Despite his tender age, the attacking winger has already earned 12 caps for his country, scoring once.
