Officers believe Mr Valiuk, whose date of birth may be 27/3/1983, was involved in a road traffic collision in Caistor on Friday, December 13.
Officers attended the collision involving a pedestrian (Mr Valiuk) and a Skoda Octavia in Audleby Top at around 9pm.
The driver of the Skoda was not injured.
A Lincolnshire Police spokesman said: “Mr Valiuk suffered serious injuries and we are urgently trying to locate his next of kin.
“Enquiries show that the man we believe to be Mr Valiuk got on to a bus in Lincoln at 10.45am on December 13.
“He travelled to Caistor and was last seen in the White Hart pub at around 4.25pm.
“He is described as white male, and he was wearing dark clothing/joggers.
“He has short dark hair and when in the pub he had a dark backpack.
“Mr Valiuk no longer had his backpack when he was involved in the collision.
“We believe he lived in the Thesiger Street area of Lincoln but he may have links to Boston and Spalding.
Detective Sergeant Alison Bowley, from the serious collisions unit, added: “Extensive enquiries have been carried out to try and identify this man who has suffered serious injuries in a collision.
“He had a bank card on him in the name of Vitalij Valiuk.
“We would like to hear from anyone who may know Vitalij Valiuk as this may help us find his family.
“We would also like to hear from anyone who may have been travelling on the bus from Lincoln to Caistor on Friday, December 13, anyone who believes they may have seen him on the bus or in Caistor and anyone who may have come across his backpack.”
• If you have any information, call 101 quoting incident reference number 442 of December 13 or email [email protected] with 442 of December 13 in the subject line of your email.
Alternatively, call Crime Stoppers anonymously on 0800 555 111.