Last Friday, a spokesperson for the Duchess said she had attempted to reach out to him in hospital but had been unable to locate him.
Mr Markle, who moved from Mexico to live in the Philippines, told the newspaper he had not received any messages to his usual phone number.
According to the PA news agency, the Duchess does not have his phone number and believes he is not in possession of his phone while in hospital.
Her team confirmed she had in fact emailed her dad, but the paper reported that he "never uses email".
On Wednesday, a spokesperson for the duchess said: "Given that a Daily Mail reporter has remained at her father's bedside throughout, broadcasting each interaction and breaching clear ethical boundaries, it has been exceedingly difficult for the Duchess to contact her father privately, despite her efforts over the past several days.
"With the support of reliable and trusted contacts, her correspondence is now safely in his hands."
Also on Wednesday, the Daily Mail's parent company DMG said in a statement that Thomas Markle had contacted their journalist Caroline Graham when he was taken ill in the Philippines and asked her to travel from LA to be with him.
"As well as being a journalist, Caroline has been friends with Mr Markle since 2018," DMG said. "She has been with Mr Markle in hospital every day, providing him with care and support.
"As soon as Mr Markle gave consent for his contact details, the name of his hospital and his room number to be provided to his daughter's spokesman, Caroline shared them.
"It is entirely untrue that each interaction has been broadcast, and the suggestion that Caroline has breached ethical boundaries is demonstrably false and vehemently denied."