The 34-year-old has yet to find a new club since his release from Aston Villa at the end of last season, despite speculation linking him with the likes of Blackpool, Hull City and Ipswich Town.
However, an increase in money on the betting markets has suggested that the former England striker could be close to sealing a move.
While his former club Leicester City remain the 5/4 favourites, Everton are now rated as a 5/1 chance. This morning the odds of him signing on at Goodison Park were as long as 16/1.
Heskey previously spent four years with Everton's rivals Liverpool up until 2004.