The 30-year-old, signed for £4million last August by then manager Roy Hodgson, played just 18 matches for the Reds after struggling to settle after his move from Fulham.
Hodgson's departure and Kenny Dalglish's arrival in early January effectively signalled the end of the left-back's Anfield career as he was immediately loaned out to Championship side Nottingham Forest.
He returned to pre-season training with the Reds but was told he was surplus to requirements and has now been moved on, for a reported fee of £1.5million, signing a three-year deal at the King Power Stadium.
Konchesky's departure is likely to be the first of several as Dalglish looks to shed some of the dead wood which has accumulated at Anfield.
Forward Milan Jovanovic, who arrived on a free transfer from Standard Liege last summer, is expected to be the next out of the door after scoring just twice in 18 matches.
The Serbia international started just one game under Dalglish and did not feature in the squad for the last three months of the season.
He may possibly return to Belgium with Anderlecht, although French champions Lille are believed to be interested in the 30-year-old.
Several other fringe players also seem set for the exit with 31-year-old midfielder Christian Poulsen and Joe Cole, 30 in November - both signings from the short-lived Hodgson era - linked with moves away.
Striker David Ngog appears to be surplus to requirements now and has been interesting some Premier League rivals, while the future of back-up goalkeeper Brad Jones appears in doubt with Liverpool apparently close to bringing in Roma's Brazilian Alexander Doni as Jose Reina's deputy.
Left-back Emiliano Insua, who spent last season on loan at Galatasaray, may have earned a reprieve, however, after he flew out to Malaysia on Tuesday to await the arrival of the first-team squad, who play their first pre-season friendly against Guangzhou in China on Wednesday.
The Argentinian did not get a visa for the China leg of the mini-tour but is hopeful of being involved against a Malaysia XI at the weekend.
"I will meet the team on Thursday. I am looking forward to playing on Saturday," he wrote on Twitter.
Leicester manager Sven-Goran Eriksson is looking forward to working with Konchesky after he penned a three-year deal with the Foxes.
"Paul has been a Premier League player year in, year out throughout his career so he represents a very big signing for us," the former England boss, who gave the defender his international debut, told the club's website.
"He wanted to come to Leicester and it means a lot that proven, established top-flight footballers want to be a part of what we are building here.
"I know Paul very well as a player and a person and he will be a top-class addition to our squad."