Susan McLean, 61, who was on a two-week tour of Scotland with her family, was reported missing from her holiday lodge in Aberfeldy, Perth and Kinross, on 17 May.
Since then extensive searches have been carried out in the surrounding area.
Human remains were found in dense undergrowth near the town at around 1.15pm on Saturday afternoon during searches carried out by volunteers.
The death is being treated as unexplained and a report will be submitted to the procurator-fiscal.
Formal identification will take place in due course, but the family of Mrs McLean has been informed of the discovery.
Superintendent Graeme Murdoch, Tayside division, said: “On Saturday 15 August, human remains were found during a volunteer search in dense undergrowth in a wooded area near Aberfeldy. “A police investigation has been ongoing following the disappearance of Susan McLean.
“The family has been informed of the discovery and formal identification will take place in due course.
“The death is being treated as unexplained and as with all sudden deaths a report will be submitted to the procurator-fiscal.
“We would like to thank members of the public who have given up their time and continued to assist the search.”
Yesterday a friend said she was “100 per cent certain” the remains found on Saturday were those of Ms McLean. Lorna VanderZanden travelled to Scotland from the US to reignite the search for Ms Mclean.
The retired US army vetinary surgeon co-ordinated another search on Saturday involving members of the public.
In a recent interview, Mrs McLean’s husband, Donald, revealed that his wife went walking on her own after the couple had a “disagreement”.
He said that she left their holiday lodge that night to “cool down” following a row between the pair.
Mr McLean, from Boiling Springs, Pennsylvania, also said that he had hired private investigators in a desperate bid to track his wife down. The last confirmed sighting of the horse riding instructor was on CCTV outside the resort at 7:45pm.
Speaking after her disappearance, he said: “Susan and I had a disagreement about dinner following our return from the Yew Pub in Fortingall.
“So, being upset, she left the lodge to go for a walk to cool off, something she often did to clear her head.”