The Reds have already seen one bid rejected for the Republic of Ireland international.
The Reds made a surprise bid of £8m for the Republic of Ireland international (via The Sun). But Ronald Koeman's men rejected the offer, as The Sun claims that they hope to keep the 28-year-old, who has scored five goals in the Premier League this season.
However, despite their desire to keep the forward, The Sun adds that a bid of around £12m may tempt the south coast club to sell.
While Long is not quite the kind of name that will capture the imagination at Anfield, it is not difficult to see why Jurgen Klopp is a fan of the striker.
Long brings a level of energy that prevents defenders from having any rest when they come up against the Saints - as he proved during the 4-0 demolition of league leaders Arsenal last month - and that would appear to suit Klopp's preferred philosophy.
And while Southampton are reluctant to accept a bid for the striker, Charlie Austin's arrival would certainly soften the blow if Long was to leave.