The 32-year-old forward, part of England's World Cup squad, appears certain to leave Southampton after a fee believed to be around £4million was agreed between the two clubs on Thursday.
Lambert spent the night at a Liverpool hotel after sitting on the bench for England's 3-0 friendly win over Peru at Wembley, and he arrived at the Spire Liverpool Hospital shortly before 9am on Saturday for a physical exam.
The player was joined by Liverpool's club doctor Zaf Iqbal before being driven to the club's Melwood training ground to continue discussions over personal terms.
Speaking to Sky Sports News at Liverpool John Lennon Airport, Lambert said: "The medical went well. There are a lot of emotions, as you can imagine. I'll speak about it properly very shortly."
Lambert, a Liverpool supporter, spent five years in the Red's youth team from the age of 10 before being released in 1997 and joining Blackpool a year later.
He joined Southampton in 2009, after spells with Macclesfield, Rochdale, Stockport and Bristol Rovers, when the Saints were a League One club under current Newcastle boss Alan Pardew.
Lambert has been the club's top goalscorer in every subsequent season since, as the club were promoted to the Championship in 2011 and then the Premier League in 2012.
The striker is expected to fly to Southampton late on Saturday before rejoining Roy Hodgson's England squad ready for their departure to the United States for the further pre-tournament friendlies against Ecuador and Honduras.
Liverpool, meanwhile, are continuing to pursue his Southampton and England teammate Adam Lallana, with a final, improved bid of £25m having been submitted to the south coast club.
There's no point looking at situations from last season because this season Liverpool are in the Champions League. I would expect him to play a lot of cup games, to be alright backup should there be injuries and also as an option in league matches where they can afford to rest players. He's a decent player, no doubt about that although hardly earth-shattering, teams need decent backup and in him (and possibly Borini) Liverpool have it.- thebigshow