The report in US celebrity gossip TMZ comes a day after doubts were cast over whether Mr Markle would attend Saturday’s royal wedding in Windsor.
He is believed to have suffered a heart attack a week ago and is in hospital in Mexico, where he lives, after experiencing serious chest pains, TMZ said.
The website said Ms Markle had tried calling him on Monday and sent him a text saying she loved him and was concerned about his health.
Mr Markle, 73, allegedly staged paparazzi photographs of himself in the run-up to the wedding at Windsor Castle, and it was reported that he had withdrawn from the ceremony as a result on Monday.
However, Mr Markle told TMZ that Meghan has said she does not have any ill feelings towards him for staging the photos.
He said that after thinking about the situation the incident was not a “serious transgression”, but simply “stupid”.
“I hate the idea of missing one of the greatest moments in history and walking my daughter down the aisle,” he told TMZ.
“This is a historic moment. I’d like to be a part of history.”
Mr Markle added: “I don’t think the Queen is thinking about what I’m doing.”
He said that if doctors allow him to be discharged, he will fly to London.
The award-winning lighting designer is hoping he will get out of hospital on Wednesday, but his doctors have refused to commit to a date, TMZ added.
Mr Markle also said his eldest daughter, Samantha, who has repeatedly spoken negatively about Ms Markle in the press, had had no real relationship with her as an adult and no knowledge of her relationship with Prince Harry.
Kensington Palace has neither confirmed or denied whether Mr Markle will attend, but has appealed for understanding.
A spokesman said: “This is a deeply personal moment for Ms Markle in the days before her wedding.
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“She and Prince Harry ask again for understanding and respect to be extended to Mr Markle in this difficult situation.”
:: Sky News will broadcast live in Windsor the week before the royal wedding with all the build-up to the big day, with exclusive live coverage of the wedding in UHD in a special programme from 9am to 3pm on Sky News and Sky One on Saturday 19 May.