The property will be made available to the couple later this year.
William, 30, who reached the milestone on Thursday, is said to be celebrating privately with family and friends over the weekend.
It is understood that he is particularly delighted about the house because he and Kate spent their first Christmas as husband and wife at Sandringham last year.
"A property is definitely being made available to William and Kate this year and a plan is in place for William to have his own place on the estate," a source was quoted by the Daily Mail as saying.
"He's a country boy and loves Norfolk, just like his father and his grandparents. It's a lovely, peaceful place and William adores it," the source added.
William is now also legally entitled to half of his mother's inheritance, which amounts to 10 million pounds, that was left to him after Princess Diana was killed in a car crash in 1997.
The 20,000-acre Sandringham estate near King's Lynn in Norfolk has been owned by the Queen since her accession in 1952 and by the Royal Family since 1862.
The Queen and the Duke of Edinburgh stay in the main house, an Edwardian mansion, every Christmas with the rest of the Royal Family.
It is not known which of the 150 residential, commercial and agricultural properties on the estate that the Queen intends to give to the Duke and Duchess of Cambridge.
Among the options are York Cottage, which is currently used as an estate office.
Built by Edward VII soon after he moved to Sandringham, it was the first home of King George V and Queen Mary after their marriage in 1893.
Sandringham was a favourite of George V, who declared it "the place I love better than anywhere in the world" and William's great-grandfather, King George VI, was born there in 1895.
While in London, the couple's official residence is in Kensington Palace in West London, Princess Diana's former home.
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