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A spokeswoman for Forbidden Corner, a folly garden created by Mr Armstrong at his 500-acre Tupgill Park estate in North Yorkshire, told the BBC they were aware of the reports from Ecuador.
But they said they did not currently have any further information about what had happened.
She said: "The thoughts and prayers of everyone at Forbidden Corner are with the Armstrong family."
According to local media in Ecuador, Mr Armstrong was kidnapped at dawn on Saturday at a farm he owned in Baba in Los Rios province.
As many as 15 men entered the property dressed as police officers and took him away, reports suggested.
It was also reported that a Colombian woman, Katherine Paola Santos, who was believed to be Mr Armstrong's partner, was kidnapped at the same time.
There were no immediate details about the captors, the AFP news agency reported.