But Martin, who has offered the 25-year-old a new deal, is keen to keep Barnard at Roots Hall despite the threat of the player being a free agent in the summer.
The Shrimpers chief told the Southend Echo: "Of course the club doesn't want to sell Lee but clearly he will not sign a new contract and I am told he will walk at the end of the season.
"In the meantime, because the player recognises that he might double or even treble his wages at the clubs that have enquired, it would be difficult for the manager to keep him motivated when the player has at least one eye on leaving and potentially at a higher level.
"There has been interest from two League One clubs over the past few days with a third club, again in League One, making enquiries. All these clubs are vying for promotion and are formally of at least Championship status."