Reports earlier this month claimed Liverpool had submitted a £20m offer for the England international, who was named in Roy Hodgson's World Cup squad on Monday.
Manchester United, Tottenham and Chelsea have all been linked with the 26-year-old as well, although Liverpool's interest is thought to be the most serious.
But Southampton, who are in no desperate need to sell, are ready to demand up to £30m to let their captain speak to interested clubs.
The report claims Lallana wants the opportunity to explore his options this season, after the best campaign of his career to date. He's helped the Saints rise from League One to the Premier League, and won four individual awards at the club's end of season presentation earlier this month, including Players' Player of the Year.
Liverpool boss Brendan Rodgers wants to strengthen his squad early this summer, as they get preparations in place for a Champions League campaign next term.
And an attacking midfielder is believed to be a priority for Rodgers, after several failed pursuits in recent transfer windows, and the Liverpool boss is a keen admirer of Lallana's creative abilities across the attacking positions.
Manchester United meanwhile are expected to complete a deal to sign Lallana's Southampton teammate, Luke Shaw.
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