The 62-year-old Israeli met officials from the club yesterday and he looks set for his 12th management post.
"I have completed the meeting with officials of Levski Sofia and I can say I am 99 per cent certain that a deal will be reached in the next days," Grant told Ghanasoccernet.com.
"There are just some few points to be ironed out but I am sure that we can complete the deal for me to coach the club with so much promise.
"I am returning to London this afternoon and I am sure I will join the club if everything goes to plan over the next few days."
Grant has been targeting a return to the dugout since stepping down as manager of Ghana in February. He led Partizan Belgrade to the Serbian League title in 2012 following spells with Chelsea, whom he led to the finals of the Champions League and League Cup, Portsmouth and West Ham.