The 29-year-old has signed a four-year contract, with Reds reserve team players Lauri Dalla Valle and Alex Kacaniklic moving in the opposite direction as part of the deal.
Konchesky was at Anfield to watch his new team's victory over West Brom on Sunday but the completion of his deal was held up by negotiations and medicals for Dalla Valle and Kacaniklic.
The two-cap former England international, who played under Hodgson at Fulham, will provide much-needed competition for Fabio Aurelio at left-back.
Argentinian Emiliano Insua is destined to leave the club, with a proposed loan to Galatasaray, after an earlier move to Fiorentina collapsed over personal terms despite the clubs agreeing a fee.
Source: PA
Source: PA