Chelsea have made a £48million bid including add-ons for Southampton midfielder Romeo Lavia.
The Championship side have been holding out for £50m and rejected a £46m offer from Liverpool - their third bid - on Tuesday.
Southampton, who have agreed to sell James Ward-Prowse to West Ham for £30m, also anticipate interest from Manchester United.
A foreign side has also made tentative enquiries for the 19-year-old Belgian.
Southampton are confident in their valuation as they believe Manchester City are willing to trigger their buy-back clause of £45m which becomes active next summer.