It is claimed Meghan, 36, tried to contact her father on Monday, but that he was away from his phone. She followed it up with a text, according to TMZ, saying she loved him and that there were no ill feelings.
He told the website: "I hate the idea of missing one of the greatest moments in history and walking my daughter down the aisle. Of course I'd walk her down the aisle. This is a historic moment. I'd like to be a part of history."
Mr Markle is said to have asked doctors to allow him to leave his bed, saying he will risk his health to fly to England before Saturday.
The American former actress's half-sister Samantha Markle said she hoped Thomas Markle would be at the ceremony, but that he had faced an "unbelievable amount of stress".
She told Good Morning Britain that Mr Markle had health issues, had had chest pains and had suffered a heart attack.
But it was not clear whether doctors had diagnosed a need for medical emergency.
"I think he wants to go to the hospital and get checked out," she said.
"I don't know as of today what his plans are, but there's a very real concern. I wanted to see him go (to the wedding), I didn't want him deprived of that.
"But clearly the priority should be whether or not it's safe for him to do that. That's where we're at."
Mr Markle became embroiled in controversy after he allegedly staged paparazzi photographs of himself in the run-up to the nuptials at Windsor Castle.
Celebrity website TMZ claimed Mr Markle had initially decided not to go to the wedding for fear of embarrassing his daughter following the fallout from the pictures.
Kensington Palace declined to confirm whether Mr Markle had pulled out of attending but appealed for understanding.
A palace spokesman said: "This is a deeply personal moment for Ms Markle in the days before her wedding.
"She and Prince Harry ask again for understanding and respect to be extended to Mr Markle in this difficult situation."
Samantha Markle defended Mr Markle, saying he had been pressurised into a dangerous situation because of press intrusion.
"He was living a very quiet, peaceful life, not as a recluse. He's a very intelligent man. He's been gregarious and worked in entertainment for 40 years," she said.
"He had spoken with press and was doing just that, but when he called me, panicking on the freeway because he's in a dangerous situation being followed by seven or eight cars, I think there have been examples in history of how dangerous that can be."
She earlier attempted to shoulder some of the blame when she tweeted that it was her idea for Mr Markle to pose for what she described as "positive photos" in a bid to combat his "bad" portrayal in the press, but insisted it was not done for money.
Pictures published earlier this month in newspapers and websites showed Mr Markle in a series of royal wedding-related activities such as being fitted for a suit, apparently unaware he was being snapped.
The Mail On Sunday newspaper reported that the images were staged and published CCTV footage stills of the US actress's father in the internet cafe with a photographer.
The former TV lighting director was due to meet the Queen and other royals with Meghan's mother, Doria Ragland, this week ahead of the wedding.
Meghan's father met her mother on the set of the US soap General Hospital, where he worked as a lighting director when she was a temp in the studio. They divorced when Meghan was six.
Ragland is being tipped to step in and accompany her daughter down the aisle if Mr Markle does not attend.
Meghan's extended family arrive in London - without invites
Meghan Markle's extended family arrived in London with seemingly no invite to the big day.
With the constant speculation over whether or not her father would attend the ceremony, her nephews' arrival with their mother heaps more embarrassment on the American actress, 36.
Tracy Dooley - who was married to Meghan Markle's half-brother Thomas Markle Jr - was pictured at Heathrow Airport with her sons, Tyler, 25, and Thomas, 26.
Ms Dooley has previously told ITV's Good Morning Britain she had not received an invitation to the ceremony at Windsor Castle on Saturday, admitting she had not seen Ms Markle for 20 years.
She is expected to cover the event for an American broadcasting network and said she "will be in the front row cheering them on" in Windsor.
The Dooleys took to Facebook to announce their presence in the capital on Monday evening.
Ms Dooley posted pictures of landmarks, including the Palace of Westminster and the London Eye, writing: "We're here. Yay."
Her son Tyler uploaded an image of the Imperial War Museum, as well as a photograph of the inside of a hotel room.
"Just touched down in London. Man, what a journey so far," he wrote. "We're safe and sound."
Ms Dooley lives in Medford, Oregon, according to her Facebook, but is originally from Calabasas in California.
She is listed as a creative advertising and marketing specialist at Opus Broadcasting - a company responsible for a handful of radio stations, according to its website.
On May 3, she shared a link to her personal blog, saying she was travelling to London to "cover all things Royal wedding".
Upon the family's arrival, the Facebook blog page - called Royal Wedding with the Dooley Markles - posted a picture of the trio landing at Heathrow with the caption: "Just glad to have landed safely and hoping for Tom senior to be well, healthy and happy.
"I wish the same for Meghan, Prince Harry, my children, friends, family and those who need our prayers and love. Peace, love and good works."
In April, the blog's page, which appears to be run by Ms Dooley, shared an article headlined "Meghan Markle snubs nephew and his mum with no invite to Royal wedding."
It was accompanied with the caption: "Beyond blessed to be any part of this momentous occasion. We'll be front and center at Windsor cheering Meghan and Prince Harry on."
This article originally appeared on the Daily Telegraph.
