Barry is now a free agent after being released by champions Manchester City this summer and has attracted interest from a number of Premier League clubs.
The 33-year-old impressed during a season-long at Everton last season but is yet to be offered a permanent stay at Goodison Park, despite reports Toffees boss Roberto Martinez wants to keep him at the club.
But Leicester are believed to be the only club to officially approach Barry over the possibility of a deal this summer, although wage demands and length of contract could prove a stumbling block for the Foxes.
Nigel Pearson has been given the backing from chairman Vichai Srivaddhanaprabha since winning promotion back to the top-flight and has already signed aston Villa’s Marc Albrighton, Man United’s Jack Barmby and Brighton’s Matthew Upson in an attempt to build a squad capable of competing in the Premier League.
Vichai has openly admitted he is willing to spend money in order to get Leicester in to the top-five of the Premier League by 2017 and signing Barry would be the first real statement of intent from the Thai billionaire.
He said earlier this month: “We want to stay in the Premier League as long as possible.
“But we won’t take the huge leap to challenge the league’s top five clubs immediately.
“I think we need to establish our foothold in the league first and then we think about our next step.
“It will take a huge amount of money, possibly 10 billion Thai Baht (around£182 million), to get there. But that doesn’t put us off. I am asking for three years, and we’ll be there.
“It will be a difficult task but we have to try to do it.”