The Reds are among a number of teams chasing the 16-year-old Fulham sensation and had hoped to sign him this summer.
But it now looks likely that Sessegnon will instead stay on and sign a new four-year deal at Craven Cottage.
Should he decide to leave, however, Liverpool would be at the front of the queue.
Liverpool are hot on the trail of £35million-rated Roma centre-back Kostas Manolas.
Boss Jurgen Klopp dispatched one of his top scouts to check on the commanding Greek defender in the Europa League against Lyon in the Italian capital on Thursday - and he was impressed.