The 32-year-old has featured on 22 occasions for Jurgen Klopp's side during the current season but is out of contract at the end of June and will not be penning an extension at Anfield.
A number of clubs are thought to be interested in the versatile attacker, who has found the back of the net on three occasions in 34 appearances at international level.
However, according to Football Insider, Leicester have now placed a 'long-term contract' on the table for the former Southampton captain as they attempt to win the race for his signature.
Foxes boss Brendan Rodgers managed Lallana at Liverpool, and it has been suggested that the pair's relationship could see the move take place this summer.
Arsenal, Tottenham Hotspur, West Ham United and Burnley are also believed to be closely monitoring developments surrounding a player who has scored 22 times and provided 22 assists in 178 outings for his current side. body check tags ::