As reported by the Daily Echo, the Reds will pay upwards of an initial £4million for the player with the possibility of further add-ons, although reports of the deal costing £9m are wide of the mark.
If everything goes to plan Lambert will travel to Merseyside after England's friendly with Peru at Wembley on Friday night and could be confirmed a Liverpool player as early as Monday.
Saints are also expected to resume talks with Manchester United today over the potential transfer of Luke Shaw.
Merseyside-born Lambert will provide some much-needed back-up to frontline strikers Luis Suarez and Daniel Sturridge and, having fought his way through the divisions to earn a World Cup call, he will relish the chance to return to the club which he left as a teenager after failing to make the grade from their academy.
Lallana was seen as the primary target to bolster the Reds' midfield, with Lovren less so for the centre of defence, but Lambert looks like being the first through the door at Anfield in what is expected to be a busy summer as they look to strengthen ahead of a return to the Champions League.