Bertrand, who used to captain the Saints, is a free agent after his contract with the club expired on 1 July.
According to a report from The Athletic, the full-back is set to sign a two-year contract with the Foxes upon completing his free transfer to the Midlands.
31-year-old Bertrand’s last game for Southampton came against Leicester in the FA Cup semi-final at Wembley back in April. Brendon Rodgers’ side went on to win that competition against Chelsea.
Bertrand first joined Southampton on loan from the aforementioned Chelsea in 2014.
That transfer was made permanent months later and the England international went to make 240 appearances for Saints in all competitions, scoring eight goals during that seven season stint.