Valerij Tomarenko was last seen getting off a ferry on the island on August 15, with his hire car found parked at the mainland terminal in Ardrossan, North Ayrshire.
A body was found on Goatfell, the highest mountain on Arran, at around 11.30am on Thursday.
Formal identification has yet to take place but the 58-year-old's family have been informed.
A police statement said: "Police Scotland can confirm the body of a man has been found on Arran.
"A formal identification has still to take place, however it's believed to be 58-year-old German tourist Valerij Tomarenko, missing since Thursday August 15. His family has been informed.
"Officers searching for Mr Tomarenko found his body on Goatfell around 11.30am on Thursday August 22.
"A report will be submitted to the procurator fiscal."