Foxes boss Sven-Goran Eriksson has enjoyed a hectic summer in the transfer market as he looks to build a squad capable of winning promotion to the Premier League.
He has already brought in a host of new signings and remains in the market for a proven goalscorer before the transfer window closes.
Eriksson saw one target Shane Long join West Brom, while Robbie Keane opted for a move to Los Angeles Galaxy after attracting the Swede's interest.
The Foxes are now said to have trained their sights on Beckford, who scored 10 goals in 38 appearances for Everton last season after joining on a free transfer from Leeds United.
Beckford grew in confidence as the season wore on at Goodison Park and it is uncertain whether Everton boss David Moyes would be prepared to sell the 27-year-old.