The Belgian midfielder has made just two Premier League starts for Spurs this term under Mauricio Pochettino and looks set for a battle to make himself a regular under the Argentine.
Roberto Martinez enquired about a potential deal over the summer, yet the Lilywhites were, at the time, keen to ensure their powerful midfielder stayed put.
But with his continued first-team woes, the Toffees are hopeful that they can bag their man in January.
The 27-year-old impressed at Fulham before his 2012 move across London, where he has struggled to match the highs he reached at Craven Cottage.
Spurs are likely to demand around £15m for their man, which could be a stumbling block for the Merseysiders – who paid £28m for Romelu Lukaku over the summer.
Dembele was an infrequent starter under Tim Sherwood last season, and is thought to have grown unsettled in that tie.
[ffcvideo file=”TopFiveVideosoftheWeek” type=”mp4″]