The 26-year-old allegedly failed to agree to an offer from Reading, believed to be worth £2,000 per week.
Adams, appointed Leicester manager back in April to replace Dave Bassett, was said to be quick onto the scene with an offer which the player immediately accepted.
The deal is expected to be announced on Monday.
Despite Royal's Chief Executive Nigel Howe denying that Reading would be making any close season signings this alleged attempt to sign Cullip indicates that there could indeed be transfer activity at the Madejski Stadium this summer.
Cullip started out at Oxford but after failing to make a first team appearance he signed for Fulham in the summer of 1996. Two years and seventy-one appearances later the 6ft 1 defender signed for Brentford in a £75,000 deal.
After an injury hit spell with the Bees Brighton came in and paid £50,000 for the player in October 1999.