Speaking on BBC Radio 4’s Today programme, Ms Cooper began to say she was speaking on behalf of the Home Affairs committee but was quickly caught out by Mr Humphrys and asked to clarify Labour’s stance.
Mr Humphrys said: “You’re also a very senior Labour MP. You used to hold government. I mean, you are a senior member of the Labour Party. Obviously the suspicion is that you’re not telling us because you don’t know.”
Ms Cooper paused and said: “No… We are conducting a second enquiry on the committee and this is across party committee. I think we are right in looking at this on a cross-party basis.”
Mr Humphrys hit back and said: “Forgive me, I am talking to you ,about it as a Labour MP, as you very well know.”
Ms Cooper continued to mention a Home Affairs committee report and how it looked at the many options for immigration after Brexit. She said: “What the Labour Party has said is they’ve said look, for the transition period, and that’s what this report focusses on, that for the transition period it should carry on the way it is at the moment.
“That actually does provide a lot more clarity.”
But the Labour MP then claimed she comes from a manufacturing constituency so would like to see Britain remain in the customs union after its exit from the Brussels bloc.
She said: “From a manufacturings point of view, whether you are in Yorkshire or the North or the Midlands – we know how important it is to be able to have a really good manufacturing deal.”
Mr Humphrys asked: “So you’d have to stay in the European Union?”
The Labour MP said: “No you wouldn’t. That’s part of the customs union deal.”
She added: “Personally, I would like us to get all the benefits and stay in the single market but I recognise that is complicated.”
But Tory MEP Dan Dalton warned Britain could be in a worse situation if it decides to remain in the customs union after Brexit – trapping it in the European Union and blocking its global trade opportunities.
Mr Dalton warned remaining in the customs union would prevent the UK from striking trade deals with third-party countries including China and South Korea, who are keen to sign agreements with Britain.
Speaking exclusively to Express.co.uk, Mr Dalton said: “It is really the worst of all worlds because you only trade with the EU.
“Given the fact that we voted to leave the EU because we wanted to look more globally and look at global opportunities that exist in third countries – that would be the absolute opposite and could be a worse situation than we have at the moment.”
Labour MPs Heidi Alexander and Alison McGovern have called on party leader Jeremy Corbyn to “get off the fence” and back remaining in the customs union and single market after Brexit.
Mr Corbyn told BBC’s Andrew Marr the UK must “have influence over the regulations” and develop a “form of customs union”. But he said the party did not want to remain in the customs union, which would require full membership of the EU.
A recent poll by YouGov suggests that Labour supporters are unclear on the party’s Brexit stance.
A quarter of people thought Labour was completely against Brexit and a third said they did not know.