It is unclear whether Meghan Markle's father Thomas will walk her down the aisle on Saturday as planned.
Markle told celebrity news website TMZ, at his home in Mexico, that he had decided not to attend his daughter's wedding to Prince Harry.
Kensington Palace issued a statement saying it was "a deeply personal moment for Ms Markle".
It follows reports in the Mail on Sunday that Markle's father staged pictures for a paparazzi photographer.
The widely circulated photographs showed him being measured for a wedding suit and looking at newspaper stories about his daughter.
He is reported to have said that he did not want to cause his daughter embarrassment by attending the wedding.
TMZ reports that Markle suffered a heart attack a week ago, although this has not been confirmed by the family or palace.
A Kensington Palace spokesman said: "This is a deeply personal moment for Markle in the days before her wedding.
"She and Prince Harry ask again for understanding and respect to be extended to Markle in this difficult situation."
Markle had been due to meet Prince Harry for the first time this week, as well as the Queen and the Duke of Edinburgh, before walking his daughter down the aisle at St George's Chapel in Windsor Castle.
BBC royal correspondent Nicholas Witchell said he understood Markle, 36, was distressed and concerned for the well-being of her father, who has been described as a rather shy and reclusive figure.
"I understand it is still very much Meghan Markle's wish that her father should be there on Saturday to support her," he added.
The controversy surrounds pictures of Markle, which were published earlier this month in a number of newspapers showing him in a series of wedding-related activities - apparently unaware he was being photographed.
Samantha Markle, Markle's half-sister from her father's first marriage, has claimed she is to blame for the photographs.
She tweeted: "The media was unfairly making him look bad so I suggested he do positive photos for his benefit and the benefit of the royal family.
"We had no idea he would be taken advantage of. It was not for money."
'Daddy's girl' Meghan
Ex-Suits actress Markle's father is a former lighting director who worked on programmes including the 80s TV show Married with Children and General Hospital, for which he and his team won two Emmy awards.
He and Markle's mother, Doria Ragland, divorced when Markle was six years old.
Markle, who has two children - including Samantha - from his first marriage, filed for bankruptcy two years ago.
His star daughter Markle has previously described him as caring, hardworking and someone who "invested in my future so that I could grow up to have so much".
She said: "It's safe to say I have always been a daddy's girl - he taught me how to fish, to appreciate Busby Berkeley films, write thank-you notes, and spend my weekends in Little Tokyo eating chicken teriyaki with vegetable tempura."
The wedding will take place on 19 May at 12:00 BST at St George's Chapel in Windsor.
Markle's parents were due to meet the Queen, Prince Charles and the Duchess of Cornwall for tea in the week leading up to the wedding.
With five days to go, Kensington Palace had earlier revealed that Markle will spend her last night before getting married at a luxury Berkshire hotel with her mother, Doria.
Prince Harry will be staying 15 miles away at the Dorchester Collection's Coworth Park in Ascot, with his brother, the Duke of Cambridge.
On Sunday, Buckingham Palace released a photo of the Queen's written consent which allows the couple to marry.