Mr Getty made the offer last week after a personal request by Timothy Clifford, director of the National Galleries of Scotland and a successful applicant for Mr Getty's wealth in the past.
But Mr Clifford undermined the gesture within 24 hours by giving broadcast interviews in which he attributed Mr Getty's generosity to a feud with his late father, Paul Getty, who died in 1976.
Paul Getty left a trust endowing the free-spending Getty Museum in California, which bought the Antonio Canova sculpture for pounds 7.6m and will be granted an export licence in three months if British museums fail to match the offer.
Mr Getty's pounds 1m donation would have left them only pounds 800,000 short, but in a letter to Mr Clifford, the American said: 'I was astonished to read the comments attributed to you suggesting that my offer of pounds 1m towards the purchase of Canova's The Three Graces was motivated by a grudge between the Getty Museum in Malibu and myself.
'This is not only untrue, but very embarrassing to myself and the museum. The only way I can see to disprove the allegation is to withdraw my offer and do everything I can to see the Getty (museum) gets it. This I propose to do unless you have any suggestions.'
Mr Clifford issued a statement yesterday offering his 'sincere apologies' to Mr Getty, and said: 'The sole motivation of Mr Getty . . . was to make a magnificent gesture to acquire a singular work of art for the United Kingdom, the country in which Mr Getty lives and for which he has the greatest affection.'
Mr Clifford later said he knew Mr Getty had received his apology, but did not expect an imminent reply. Mr Getty was unavailable for comment.
Copies of the letter to Mr Clifford were sent to the Department of National Heritage, the Victoria and Albert Museum and to John Walsh, director of the Getty Museum, who is understood to have leaked it to the press. No one at the museum was prepared to comment last night. Stephen Dorrell, Secretary of State for National Heritage, also declined to intervene in the controversy before flying to the Commonwealth Games in Canada.
John Paul Getty, 61, is still a US citizen but has lived in England since 1971, and was given an honorary knighthood in 1986 for his donations to the arts. He has pitted his chequebook against the wealth of the Californian museum created by his father. In 1984, he reportedly gave more than pounds 100,000 to stop a painting by 13th-century Italian artist Duccio di Buoninsegna leaving Manchester City Art Gallery for California. The gallery's director then was Timothy Clifford.
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