Liverpool have also been heavily linked with a £8million move for Hull City left-back Andy Robertson, but the ECHO understands that is unlikely to materialise.
The Reds have scouted the Scotland international previously but they currently have no plans to follow up their interest.
There are a number of names ahead of Robertson and one of them is Sessegnon, who has enjoyed an outstanding breakthrough season in the Championship.
Fulham are reluctant to sell but their hopes of keeping him were dented by their play-off exit at the hands of Reading on Tuesday night.
Promotion to the Premier League may have convinced Sessegnon to stay put for a further 12 months, but now he will have a number of lucrative offers to make the leap into the top-flight.