The 31-year-old - a boyhood Liverpool fan and former Anfield trainee - has returned to Merseyside after five fine seasons on the English south coast.
Lambert has enjoyed a prolific rise through the divisions with Southampton, scoring 115 goals in 229 games since signing from Bristol Rovers in 2009.
He remained a pivotal figure in the top flight at St Mary's Stadium under managers Nigel Adkins and Mauricio Pochettino, helping them to a record Premier League points tally last season.
His exit continues what looks like becoming a string of departures from St Mary's this close-season, with Pochettino already having left for Tottenham.
Southampton captain Adam Lallana is also expected to leave the club, with Liverpool reportedly in pole position to bring in the playmaker.
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