The Duchess’ 58-year-old brother launched the bid for Thomas Senior, 78, to be put under the legal arrangement while he recovers from a stroke.
The move, if successful, will mean Thomas Jr will oversee all of his father’s financial affairs and make decisions on his behalf.
It will also mean he will manage a court case between his dad and British photographer Jeff Rayner.
Thomas Markle is suing Mr Rayner over staged photographs taken before Harry and Meghan’s wedding in 2018.
Mr Markle was caught on CCTV posing for faked photographs taken by Mr Rayner, which were then sold for a reported £100,000 to media outlets.
The photos show him being measured for a suit to wear to the wedding, browsing a book of British landmarks and working out with weights to get in shape ahead of the big day.
The images are said to have led to Thomas and Meghan’s relationship breakdown.
She has reportedly not spoken to her dad since the pictures were released and he also missed her wedding after suffering from a heart attack.
Thomas Markle Snr has also never met Prince Harry or his two grandchildren, Archie and Lilibet.
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Information about the conservatorship arrangement was revealed in court documents filed in the Rayner case.
Meghan’s half-brother told the court he was “in the process” of obtaining the conservatorship for his dad.
A well-placed source told the Sunday Mirror: “Tom Snr has had a series of health scares and his son wants to do the best for him by taking any stress away.
“Cases in America grind on for months or years so it will be a weight off Tom Snr’s mind. It will help him focus on his recovery.”
Mr Rayner applied for a restraining order against Meghan’s dad after he was quoted saying he would consider killing him.
However, he has since denied that he said it.
In Tom Bower’s book, Revenge: Meghan , Harry and the War Between The Windsors, he said: “I got screwed Rayner… I want to tear him down. If they tell me I’ve got terminal cancer, then I’ll kill him because I have nothing to lose.”
He is set to appear in court in November, as he seeks $1million (£842, 395) claiming Rayner’s staged photos caused him “worldwide embarrassment”.