Artur Rej was last seen on January 7 in Harehills.
Officers have been carrying out "extensive enquiries" to try and track down Artur but have so far failed to do so. West Yorkshire Police are now appealing for the local community to come forward with any information they may have on where Artur is.
The 41-year-old, who is originally from Lithuania, is from the Sandhurst Road area of Harehills.
Detective Inspector Ian Thornes said: “We have been carrying out extensive enquiries into Artur’s whereabouts and are growing increasingly concerned for his welfare.
“Artur is known in the local community and within the Lithuanian community in Harehills. I would appeal for anyone who saw Artur around the 7 January or since the 7 January to contact officers.
“I would also appeal for anyone with information on Artur’s whereabouts to come forward.”
Anyone with information should contact police on 101, quoting reference 13200105936.