He set off from the village of Aegiali, in the north of the island, and was heading for the town of Katapola, the main port. The hike takes around four hours under normal conditions.
It is a well-known and well-maintained trail.
When he failed to turn up, a friend reported him as missing. The island is subject to a “severe high temperature warning” this week.
A search operation has been launched, with additional emergency personnel arriving from the nearby island of Naxos.
The American is well known on Amorgos, having visited several times.
“He had completed the Aegiali to Katapola hike several times in the past,” Calliope Despotidi, the deputy mayor of Amorgos, told Greek Reporter, a news website.
“It is strange because he is not a person who walked the route for the first time. He knows Amorgos better than me.”
She said she was concerned that scorching temperatures may have been a factor in his failure to turn up at his destination.
“He may have lost consciousness because of the heat and hopefully found shade in one of the caves along the route,” Ms Despotidi said.
She said she hoped the American hiker had not met the same fate as Dr Mosley, 67. His body was found among rocks on the coast of Symi on Sunday, four days after he was reported missing.