Frank Lampard’s side are looking to do further business before the transfer window slams shut and may need to replace winger Anthony Gordon, who is wanted by Chelsea.
Everton have Januzaj, who is a free agent following his exit from Real Sociedad, on their radar, according to Belgian journalist Sacha Tavolieri, and the winger is keen.
Januzaj is aware that he is under consideration at Goodison Park and is waiting for Everton to make an approach.
The former Manchester United star would be keen to join Everton as his priority this summer is to get back to the Premier League.
Januzaj, who is now 27 years old, made 33 appearances in La Liga for Real Sociedad over the course of last season, scoring three goals.
He turned out 63 times during his period on the books at Manchester United.
It is unclear how long Januzaj is ready to hold on for Everton, but as a free agent he could join a new club even after the transfer window has closed.
