Karl Law, 34, from Fareham, Hampshire, did not return from a night out during his cousin Lee Cross's stag do in the Czech capital on Saturday 15 November.
A statement released by Czech police said there was evidence Mr Law had "consumed narcotic drugs and psychotropic substances extensively".
The claims have been denied by Mr Cross.
He said: "Obviously, Karl would have been under the influence of drink, but I've never seen him take drugs."
Friends and family who have remained in Prague to search for Mr Law, a builder, said they had now hired a private investigator.
They were also due to make a second appeal on Czech television station TV Nova to find him.
Police have confirmed two new sightings of Mr Law, on Sunday 16 November at the Palacky Square and the following day in the area of the National Theatre.
Prague police spokesman Tomas Hulan said: "In both of the cases he allegedly showed signs of intoxication of alcohol and some confusion."
Police have denied allegations that officers did not co-operate with the family in the search for Mr Law.
The family, who are walking the city centre and its outskirts wearing T-shirts printed with Mr Law's picture and handing out leaflets, said they had noticed police had "upped" their search.
Mr Cross said: "I'm just hoping and praying that he's OK; it's about time we found him and took him home.
"I don't want to leave here without him."
Hampshire police has confirmed its officers are in contact with counterparts in Prague and are helping to follow-up lines of inquiry in the UK.
Chief Inspector Jim Pegler, policing commander for Fareham, said: "We recognise that this is a very difficult and frustrating time for Karl's family and friends and will do all we can to assist them."
A Foreign and Commonwealth Office spokesman added: "We can confirm a British national has been reported missing in Prague. We are in close contact with the family and are providing consular assistance."
Friends and family in Fareham have held fundraisers to support the search for Mr Law in the Czech capital.
A man who went missing on a stag weekend in Prague may have taken drugs, Czech police have said.