Carwyn Jones, who claims a hard Brexit is not the right option, warned Mr Corbyn that leaving the customs union would be “economically daft”.
But Labour’s position remains unclear with Mr Corbyn suggesting the party would want a “form of customs union” despite backbenchers proposing to keep Britain in the EU club.
Express.co.uk asked Labour if the party wanted Britain to remain in the EU Customs Union.
A Labour spokesman said: “We want a final deal that retains the benefits of the Customs Union.
“Options for achieving the benefits of the Customs Union should not be swept off the table.”
The statement comes as Welsh First Minister Mr Jones said he expects Labour’s policy will change “within months”.
He told the Independent: “Jeremy is smart enough to understand that this is a debate that has to be looked at afresh from time to time, but above all we have to be guided by what is best for the people of Britain.
“A hard Brexit, which means putting barriers up between ourselves and our biggest market, is not the right option.”
Suggesting that the party leadership could bow to mounting calls for a shift, he added: “Keir Starmer, Jeremy and I all understand that we’ve got to find the best way forward for working people.”
Meanwhile, Sir Keir Starmer said he “wouldn’t rule out” staying in the customs union post Brexit.
The shadow Brexit secretary said Labour would want to leave that option “on the table” after branding the Government’s Brexit approach an “embarrassment”.
Theresa May’s Brexit ‘war Cabinet’ have been meeting this week in a bid to establish a set of agreed policies on Britain’s future relationship with the EU and Labour’s top team have also been holding meetings to try and work out their own Brexit goals.
Last month, Mr Corbyn told the BBC’s Andrew Marr that Labour “would want a form of customs union”, but added Britain “would need to be sure that any agreement with the EU gives us the chance to influence the situation we are in and the trading relationships we want”.
In December, Mr Corbyn suffered a widespread rebellion form the Labour backbenchers over an EU Withdrawal Bill amendment to keep Britain in the customs union.
The vote to amend the EU Withdrawal Bill was defeated as expected – but 64 Labour MPs defied the party’s whip, which instructed them to abstain.
Rebecca Long-Bailey, the shadow business secretary, has said it was “certainly a possibility” that Labour’s proposal to keep the UK in a customs union with the EU could block the Government from agreeing deals with other countries.
Asked if that would stop the UK from signing trade deals with other nations, Ms Long-Bailey said: “If it was a separate customs union to the current customs union for example that would be a matter for negotiation but it still has to be remembered that you would have to abide by the rules of that customs union.
“You are negotiating as a bloc – that is the whole point of a customs union.”