The devolved Scottish Government is intent on pursuing the measure, despite European opposition, and Northern Ireland is reviewing its policy.
David Cameron had pledged to enforce a 45p minimum unit price on alcohol in England and Wales, but a Cabinet revolt and public criticism of the scheme from consumers and the drinks industry sparked a U-turn.
But now Welsh Health Minister Mark Drakeford said he is seriously considering whether Wales could go it alone.
He also wants to explore the possibility of introducing plain packaging on cigarettes in Wales – another measure abandoned by the UK Government.
He said: “I was very disappointed that, on a UK level, plans which we thought were coming along have not been included in the UK Government’s programme.
“I still think that moving together is the best way to deal with some of these big public health issues. But I have asked my officials and the Chief Medical Officer to give me advice on what our powers are and if it is possible for us to move ahead with that.
“I do not align myself with the anxieties of those who accuse governments of being a nanny state. There are larger public interest issues at stake.
“Neither area is straightforward as far as legal powers are concerned, but the Welsh Government continues to explore both as we develop proposals to take forward the public health agenda.”
Since minimum unit pricing and plain packaging plans were scrapped various organisations have walked away from the Responsibility Deal between the Government and the alcohol industry that sought to foster a culture of responsible drinking in the UK.
The Faculty of Public Health, Cancer Research UK and the UK Health Forum are among the organisations to walk away from the deal in protest of the Government’s U-turns, leaving it mainly made up of representatives from the food and drink industry.
Birmingham City Council, the UK’s largest local authority, has also quit the Responsibility Deal, and said it will come up with its own plans to combat alcohol abuse, smoking and obesity.
Cllr Steve Bedser, cabinet member for health and wellbeing, said: “We’re withdrawing from the deal because we feel the Government has already broken the deal and is not acting responsibly.
“The two things that are driving this are our concerns about the Government’s failure to implement plain packaging and the failure to tackle cheap super strength alcohol. These failures will lead to more lives tragically being lost in the city.”
Dr Adrian Phillips, Birmingham’s director of public health, added: “I am really disappointed with the lack of progress with the responsibility deal across the country.
“Birmingham council signed up to the initiative when it was first announced but we now believe that there is no alternative but to try a different approach because it clearly hasn’t worked.”
The deal, signed in 2011, did not include minimum unit pricing or plain packaging.