Kent Police are appealing for information about Dominic Madar, aged 27, who left his home in the town on January 14.
Mr Madar is known to have stayed in Vienna and Bratislava, in Slovakia, and was also intending to travel to Budapest.
He was last seen on February 1, when he went to the British Embassy in Bratislava and was believed to be without any money or a phone.
On the same day he is known to have contacted someone in Budapest asking for a place to stay.
Since then Mr Madar has not been seen or heard from and it is unclear whether he made the journey to the Hungarian capital.
There are growing concerns for his welfare and Kent Police have liaised with a number of foreign embassies as part of ongoing enquiries to locate him.
Appeals for information have been circulated in embassies and hostels at locations where he may have stayed.
Officers are now urging friends or people who may have met Mr Madar on his travels to get in touch if they have any information about him.
He is described as white, around 6ft tall, of slim build with dark brown hair.
If you can help call 101, quoting reference KP-20190204-1266.