Brendan Rodgers has tabled two bids for the England international - the last a take-it-or-leave-it £25m - with Liverpool hopeful they would be able to move the deal forward once Saints announced Ronald Koeman as their new manager.
But the Dutchman's arrival does not appear to have altered Southampton's stance, with executive director Les Reed adamant Lallana, or any other player, will not leave St Mary's unless it makes financial sense for the south coast club.
"Lallana's been here long enough, we've brought him to this level and we feel that perhaps he should give it a little bit longer," he told BBC South Today.
"Anybody seriously considering coming for our players is going to have to think very significantly about the size of the fee.
"If we do take money it will be because it's the right amount and it enables us to improve the team and move forward."
Southampton are holding out for more money for Lallana as they owe former club Bournemouth a 25 per cent sell on fee.