John Joyce, 72, went missing while on a religious pilgrimage to the country on July 14 and there has been no news or sightings since then - with his family making a fresh appeal for information about his whereabouts.
But they have been forced to concede that something "darker" and "sinister" may have happened to him.
Cathie Joyce told religious news website Premier that "his family and friends are very, very worried and shocked".
Mr Joyce went missing three days into a six day pilgrimage to Fatima.
On July 14 at 9:20am, he walked with his pilgrimage group to a studio nearby to see a video about the Children of Fatima.
It finished at 10:45 am and one of the group saw him in front of the Basilica about ten minutes later, where he was last captured on CCTV in the Portuguese town.
His sister told the website that he had "left his passport, money, mobile, reading glasses, hat…in his hotel, so was obviously not intending to go far before returning to the hotel for lunch at 1pm."
She added: "If he had collapsed, fallen or been taken ill in town he would have been seen and cared for. Fatima was full of pilgrims and visitors that day.
"If he had strolled out of town and been taken ill or fallen he would have been found by now.
"The family has now been forced to contemplate 'darker' scenarios.
"Was John either beguiled into accepting a car ride or taken by force to somewhere else? Is he now being held against his will by a group or cult or for some other purpose eg. organ removal?"
The family are appealing to anyone going to Fatima to be on the lookout for John who they say may now have a beard and "may be very unkempt looking and dirty".
Mr Joyce is missing his lower denture and speaks a little Portuguese but with an English accent.
Local police and the British Embassy in Portugal are helping in the hunt for Mr Joyce.
Anyone with information can contact the Lucie Blackman Trust on 0800 098 8485, email any information to [email protected], or contact the local Fátima GNR police on (+351) 249 530 580.
He is described as being of large build and about 5ft 6in in height.
His hair is a crewcut style, balding and he was wearing black mid-calf shorts, a T-shirt and black sandals.