Nathaniel could take on Frankel in the Juddmonte International Stakes at York following his narrow King George defeat.
The John Gosden-trained four-year-old, who won the Coral-Eclipse only two weeks earlier, went down by a nose to Arc winner Danedream.
Nathaniel could renew rivalry with his Ascot nemesis at Longchamp in October, but might have a more pressing engagement against the world's highest-rated thoroughbred.
"He ate up well and is fine," said Gosden. "The great thing is that he doesn't know he got beat. He thinks it was a dead-heat!
"The races we are looking at for him are the Juddmonte International, the Arc, and the Champion Stakes at Leopardstown and Newmarket.
"He will probably run in two of those."
Update:
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