Euan Stevenson, from the Borders, was last seen on Wednesday, March 4.
He was spotted in Bay Street, Port Melbourne, Australia, and purchased a bottle of water and a cooler bag using his phone.
His family have since sent out an urgent appeal to find as they are growing more concerned for his welfare.
Euan does not have his bank card or passport on him, and his phone has been switched off.
His sister, Joanne, said: “Police have confirmed that Euan was last sighted one hour after he went missing on Wednesday, March 4, at 10.23am at Coles on Bay Street, Port Melbourne and purchased a bottle of water and a cooler bag.
“He used his phone to make this purchase, via mobile app, and this was the last transaction made on his account.
“Euan, if you see this, please make contact with anyone and know that we love and care for you so much, we just want to know you are safe.”
It is believed Euan was last seen wearing black joggers, black trainers, a white t-shirt and a pink cap.
A Victoria Police spokesman said: “Police and family have concerns for Euan as it out of character for him not to stay in touch.“Investigators have released an image of Euan in the hope someone may have seen him and can provide information regarding his whereabouts.
“Euan is Caucasian in appearance, of slim build and speaks with a Scottish accent.
“Anyone with information about Euan is urged to contact Melbourne West Police Station on (03) 8690 4444.”