Liverpool and Southampton held further talks on Thursday over a proposed £45million deal for midfielder Romeo Lavia.
Mail Sport reported earlier this week that the club were preparing a revised bid shy of £50m, Lavia's price tag, after their opening offer of £34m plus £4m was rejected.
Jurgen Klopp is believed to be keen to bring the talented teenager to Anfield this summer and an improved offer may tempt Saints into a reluctant sale.
Lavia, who moved to St Mary's from Manchester City last year, is in demand ahead of the new season after impressing during his debut campaign on the south coast.