Saints will return with a second offer for Tottenham winger Andros Townsend, despite having had a £10m bid knocked back for the player, according to the Daily Mirror.
However, the paper says that Spurs are unlikely to do business, due to Saints' refusal to sell either Morgan Schneiderlin or Jay Rodriguez to them.
The Mirror claims that, if Ronald Koeman wants to sign the 23-year-old, he may have to relax his stance on not selling Schneiderlin.
Meanwhile, reports in the Italian press suggest that Saints remain keen on Lazio midfielder Alvaro Gonzalez.
The 29-year-old Uruguay international can play out wide, in central midfield, or even as a full-back.
Saints have been linked with an £8m move for the player.
Elsewhere, The Guardian say that Malky Mackay will be appointed new manager of Crystal Palace and will resume the club's interest in Saints midfielder Jack Cork, who has a year left to run on his contract.