The glamour model is said to have agreed to record the footage about her split for an ITV world exclusive which will air on July 11, the Daily Star reports.
An ITV insider said: "Piers is over the moon to get the Jordan exclusive. Jonathan Ross will be green with envy that she turned him down and went with his biggest rival."
Price reportedly signed the deal with Morgan on Monday at the Dorchester Hotel in London and the show will be filmed in the next few days.
Morgan interviewed Price earlier this year for his Life Stories programme, during which she revealed that she was sexually abused at the age of six.