Lallana came through the Southampton academy, which also produced Gareth Bale, Theo Walcott and Alex Oxlade-Chamberlain, and has become an integral member of a side that achieved its highest Premier League points total this season.
Southampton must pay 25% of any transfer fee for Lallana to Bournemouth, who sold the midfielder to his current club.
Liverpool want to complete the deal swiftly or they will consider moving on to alternative transfer targets but with Saints captain Lallana is due to fly to the United States with England on Sunday ahead of next month's World Cup time is tight.
The 26-year-old played 73 minutes of England's 3-0 friendly win over Peru at Wembley on Friday, his fourth senior appearance for his country.
The Saints turned down Liverpool's opening offer of £20m after indicating that they wanted to end up with that amount from any deal themselves.
Liverpool's latest bid would leave Southampton with £18.75m with the remaining £6.25m going to Bournemouth.
Southampton and England striker Rickie Lambert will have a medical at Liverpool on Saturday after the clubs agreed a fee of about £4m plus add-ons.
Lambert will also be part of the England World Cup squad that leaves on Sunday for friendlies against Ecuador and Honduras.