He said the relics should not have been put up for sale as they had been stolen from Beijing's Summer Palace, which was razed in 1860 by French and British forces.
"It was just that the opportunity came to me. I was merely fulfilling my responsibilities," he said, adding "I must stress that this money I cannot pay".
Cai's foundation, formally called the China Fund for Recovering Cultural Artefacts Lost Overseas, says on its website that it was set up in 2002 by a group of academics and "prominent people".
|The bronzes were taken from China during the Second Opium War [AFP]
A spokesman for the China's State Administration of Cultural Heritage told Reuters the foundation was not affiliated to the government.
Cai's statement did not specify whether he would not pay for the relics because he did not have the money, or whether his inability to pay was for other reasons, such as his conscience not allowing him to buy looted items.
He had previously paid $14.95m for a Ming dynasty Buddha image at a Sotheby's auction in Hong Kong in 2006.
Wang Weiming, one of the heads of the artefacts recovery foundation, said he was "not sure" if or when the bronzes would return to China.
"These national treasures are probably still in France," Wang told Reuters. "We'll have to see how the situation develops."
'Threat to business'
Last week the Chinese government called on auctioneers Christie's to withdraw the two bronzes from sale, saying they were part of China's looted cultural heritage.
Christie's refused and the auction went ahead, leading the government to state that the sale had "harmed the cultural rights and hurt the feelings of China's people".
The State Administration of Cultural Heritage also warned Christie's the sale would "seriously impact" the auctioneer's business in China.
"[The agency] resolutely opposes and condemns all auctions of artefacts illegally taken abroad. Christie's must take responsibility for the consequences created by this auction," it said.