The Reds made an enquiry for Sessegnon in January as a host of leading Premier League clubs circle the 16-year-old, including both Manchester clubs, as well as Arsenal and Tottenham.
Liverpool, Manchester City and Spurs have expressed the strongest interest in Sessegnon to date for the highly rated left-back, who can also play as a winger.
Sessegnon will make a careful decision over his future as he considers the best destination for his development, but Fulham’s demands for £15m could prove to be a barrier to any deal.
Jurgen Klopp is in the market for at least one new left-back in the summer having played James Milner out of position in the role this season, with Alberto Moreno largely named as a substitute.
Sessegnon does not turn 17 until May – when his first professional contract will be activated – but is expected to be sold at the end of the season despite Fulham’s valuation.
The youngster has been eased into the Fulham side this season by manager Slavisa Jokanovic, making 17 appearances in all competitions, scoring four goals and providing two assists.
Fulham would be keen for a loan-back deal to keep Sessegnon next season, but both Liverpool and Spurs would be keen to retain the services of the England under-17 international.
Sessegnon joined Fulham at the age of nine but grew up as a Liverpool fan and the Merseyside club are eager to snap up one of the best young players in the country.
They are aware that they face stiff competition for Sessegnon’s signature, though. They are wary of missing out on him in the same way Tottenham signed Dele Alli from MK Dons from under their noses in January 2015 before another Reds supporter, Patrick Roberts, left Fulham for Manchester City for £12m six months later.