Press Association Sport understands discussions are ongoing with the Saints over the possibility of a move but they are not yet close to agreeing a fee and no conclusion is likely in the near future.
The Reds, who are also interested in Lallana's team-mate Dejan Lovren, are keen to take the 26-year-old England international to Anfield but are considering alternatives should they reach a sticking point.
It is understood the right-back, who turns 30 in August, has been offered the opportunity to stay for an additional 12 months but the England international is looking for a longer-term contract.
Johnson is one of the highest earners in the squad and the club do not want to commit significant finances to a lengthier deal and are content to allow his contract to run down should no agreement by reached.
There is no likelihood, however, that fellow England full-back Ashley Cole will be linking up with his international team-mate at Anfield after the 33-year-old admitted on Monday he is unlikely to stay at Chelsea following the expiration of his contract and is considering his options.
Liverpool have an interest in Swansea goalkeeper Michel Vorm to provide competition for Simon Mignolet while they are also tracking Bayer Leverkusen's Germany Under-21 midfielder Emre Can, although any approach is complicated by the buy-back clause held by former club Bayern Munich.
The club are putting the final touches to Rodgers' own contract, however, after the Northern Irishman said he would sign a new, improved deal once the season was over having guided the club to second in the Premier League.
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