Markle moved to the Philippines last year after he had a series of strokes.
He is reportedly now living in a Florida rehabilitation centre learning to walk again after having a leg amputated in 2025.
Meghan's half-brother Thomas Markle Jnr claimed he found a stash of images belonging to his father at his former Philippines home, the Sun reported.
He is said to have told police he discovered child abuse videos and pictures among the items. He is understood to have contacted the FBI in July.
In a statement released by Meghan on Friday she said that when she received the information of the claims against her father she was "given assurance that these were being investigated".
Meghan said she hoped it could be understood why she had kept her distance from her father and did "not allow him to have contact with our children".
Authorities in the Philippines have confirmed they are investigating the allegations about Meghan's father.
The BBC was told on Friday the matter was being investigated by an anti-human trafficking and child sexual exploitation task force.
Philippines Interior Minister Jonvic Remulla separately said national police had secured his computer and would carry out a digital forensic examination.
Meghan is the only daughter of Markle and her mother Doria Ragland.
He pulled out of attending Meghan and Harry's wedding in 2018 - citing ill health - and they became estranged.
The duke and duchess moved to the US in 2020 after stepping down as working royals.
They returned to the UK with their children this year after six years away.
Source: BBC News · Dan Sales
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