It comes as the Saints captain and the rest of the England squad set off for Miami this afternoon.
The 26-year-old was picture on the plane sitting next to midfield Jack Wilshere on which the Arsenal player posted on the social network site Instagram, below.
Wilshere posted: "Me and Mr Lallana ready for a long flight!"
Liverpool had initially offered £20m for the 26-year-old, but were unable to convince Saints to sell for that amount.
Yesterday it was reported that they have now upped that offer to £25 million in a bid to convince Saints to sell.
The Anfield side are understood to have said that their latest approach is a final one, and have threatened to move onto other targets if the transfer is not concluded before the World Cup.
That is the same tactic they used to help push through a deal for Lallana's club and international teammate, Rickie Lambert, who is expected to complete his switch to Anfield on Monday.
But Saints look likely to hold off on any offer for Lallana before the World Cup.
The club feel they are in a stronger position when it comes to Lallana, and also Luke Shaw (below), with Manchester United making their own final push to sign the left-back before England depart for Brazil.
United have already offered £27m for the 18-year-old and are keen to wrap up a deal before anyone else can swoop in and hijack it, approaching Saints again yesterday to try and move things forward.
The club want to put transfer dealings on hold until a new manager is in place and the World Cup is over.
It is increasingly unlikely that either player will remain at the club this summer, but Saints feel under less pressure to deal with those matters quickly than they did with Lambert.
Even if they do feel a sale is inevitable, the club know that interest will not disappear and their values may even rise in Brazil.
When they reach Florida Roy Hodgson's squad will take part in a pre-World Cup training camp, before jetting off to South America next Sunday.
Hodgson has said he will allow clubs to send out staff to Florida to conduct medicals in the players' free time if absolutely necessary, but nothing will be allowed once England head to Brazil.
That leaves just a week for any transfers to be completed, although even once the squad depart today it will become increasingly difficult.