Valencia had decided against activating a €12m (£10.4m, $13.4m) clause after 12-month loan.
Reds' record European appearance-maker optimistic ahead of the summer.
Virgil van Dijk is contracted to Southampton until the summer of 2022.
Blues also monitoring Roma's Antonio Rudiger, according to reports.
Bertrand, 27, reportedly wants to quit St Mary's in the summer.
Van Dijk has long been a transfer target for Liverpool, Manchester City and Chelsea.
Van Dijk remains Liverpool's number one summer transfer target, according to reports.
Dutch defender continues to be linked with move away from St Mary's.
City are willing to pay more than their rivals to land 25-year-old centre-back, according to reports.
Reds set to go on big summer spending spree in preparation for Champions League football next season.
Klopp eager to bolster squad ahead of Champions League campaign.
Belgium international goes off with undisclosed injury during goalless draw at Southampton.
Sergio Romero saves Manolo Gabbiadini penalty as Saints hold Europa League final-bound United.
Pogba is currently in France after learning of the death of his father.
Saints captain back running as he continues to recover from serious ankle injury.
Browne announced his decision to come out of retirement earlier this year.
Liverpool drew 0-0 with Southampton at Anfield last weekend.
Romeu has committed to the Saints until 2021 after signing an extension in January.
Sanchez and Giroud send Gunners fifth with 2-0 win over Southampton.
Alexis Sanchez and Olivier Giroud goals send Arsenal three points off the Champions League places.
Van Dijk has not played since 22 January due to an ankle injury.
Spanish forward Lucas Perez now fully fit after returning to training on Monday.