The start date is planned for April 1, after concerns were initially raised last year that work might not begin until 2021.
Radioactive particles were first discovered at the headland near Dalgety Bay Sailing Club as early as 1990.
Previous studies of the coastline suggest such incinerated radioactive waste was dumped prior to 1959, when the nearby airbase HMS Merlin was decommissioned.
After years of refusing to accept liabilitys, the MOD was named as the polluter by Scottish Environmental Protection Agency (SEPA).
Stephen Ritchie, from the MOD has said it is ready to put the contract out to tender over this summer.
At the south and west Fife area committee on Wednesday, Mr Ritchie said this should allow for an April start date.
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He added: “It is out to tender at the moment, although the money has not been approved yet. After that, it still has to go back to a committee for approval with the business case.
“I am more than 50% content that we will get it, but I can’t guarantee. It is out with my control.”
The contract bids are expected to come back in July which will allow the MOD to assess who is best suited to be issued a contract to clean up the waste.
Councillor David Barratt asked: “Given the time scales, what would be the final cut off point to say it wouldn’t go forward next year?”
Mr Ritchie said there wouldn’t be a final cut off date, but did agree if they missed the window for a spring start date, more costs would be involved which could cause delays.
He added: “The contract award date is expected in the Christmas holidays, so that gives three months between the contract and the first spade in the ground, to get everything in place.”
Convener of the committee Alice McGarry asked: “What if there are no contractors able to carry out the work?”
Mr Ritchie said: “We’re going through the process and will have to just wait and see what happens. The worst case scenario would be that no one comes in with a bid, and honestly I don’t know what we’re going to do if that happens.
“However, they could all come back with bids and we have to go through them and see who is best suited. Until they come back, there is nothing we can do.”
Mr Ritchie offered to come back to the committee in August to update on the progress made with bids.