Morton, a long-time royal watcher, made the comments to TV channel Telecinco's El Gran Debate programme on Saturday.
"For a little while there was a talk of a relationship between them (Felipe and Paltrow)," Morton said during the show.
"But the rumour didn't last long."
In the interview, Morton, who is in the midst of a promotional campaign for his new book about Spain's royal women, also said the abdication of King Juan Carlos could be imminent.
"It's possible it will happen sooner than we think," Morton said.
"He may do it (step down) when the scandal with Iñaki Urdangarin has been cleared up," said the royal watcher, referring to a corruption scandal that involves the king's son-in-law's alleged illegal activities at a charitable sports foundation.
"Or the king could do it on the 40th anniversary of the restoration of the monarchy (after the death of the dictator Franco) in 2015."
"Prince Felipe could assume the role (of King) before we expect," Morton concluded.
In the interview, Morton also spoke about King Juan Carlos' alleged relationship with the German princess Corinna Sayn-Wittgenstein.
Spanish newspapers have described the princess as "a special friend" of the King.
"There's a rumour that the King invited his children to dinner and then asked them to make his relationship with Corinna official.
"That's ridiculous. If the King wants to do something, he doesn't ask permission," said Morton. "He just does it."
Morton also dismissed as "ridiculous" the suggestion that the King had notched up more than 1,500 lovers.
Morton said the German princess had decided she didn't want to be the focus of all the scandals around the Bourbon monarchy.
"She has decided to give her point of view, with or without the permission of the King because she's tired of the rumours," said the royal watcher.
Corinna Sayn-Wittgenstein recently told Spanish newspaper El Mundo that she was not romantically involved with King Juan Carlos.
During his interview with Telecinco, Morton also spoke more generally about Crown Prince Felipe's romantic history.
Morton said the relationship with Letizia had been positive for Spain's heir to the throne.
"Those I've spoken to in the circle of Prince Felipe say that he is more centred, calmer and happier," said the royal commentator.
"His speeches are better, and he manages himself better in public," he added.
"Whether we like it or not, she's going to be the future queen of Spain.
"The weight of the crown rests on her slender shoulders.
"She is the motor behind the prince," Morton said.
In a recent interview with El Mundo, the royal expert also said the Spanish monarchy could learn something from their British counterparts.
They should "keep calm and carry on", Morton told the newspaper.
He also said the King's visit to India last year to promote Spain was a good example of how Spain's royal family could show it was still useful and relevant.
Asked whether the royals could survive the current economic crisis, Morton said this was a decision for Spanish politicians only but that he saw no evidence that the country's political class wanted to eradicate the institution.
Morton said from this perspective the "future of the Bourbon family was assured".
Andrew Morton has just released a tell-all book called Ladies of Spain in which he looks at the lives of Spain's Queen Sofia, the queen's daughters Elena and Cristina, and the queen's daughter-in-law Letizia.
The book is not yet available in English.
Morton is best known as the author of 'Diana: her true story', a biography of the late Princess of Wales that precipitated the public break-up of her marriage.
The book later turned out to have been based on extensive secret interviews that Morton carried out with the princess.