Dr Fox has written to colleagues setting out four options for devolved governments’ role in negotiating free trade agreements after the UK leaves the European Union, a Whitehall source confirmed.
One of them includes making trade a reserved matter for the UK Government, although at the other end of the spectrum is a proposal that a common position should be agreed with devolved governments before striking a deal.
Any move to freeze out devolved governments is likely to be strongly opposed in Edinburgh and Cardiff.
The Government has not taken a decision on which option it prefers.
However The Times claimed Dr Fox favours denying Scotland and Wales a veto, and Tories worried about the anti-Brexit Scottish National Party scuppering any free trade deals could back him.
Genetically modified (GM) foods - which are legal for cultivation in England and the United States but banned in Scotland, Wales and Northern Ireland - is one potential flashpoint in trade talks.
The Government is set to publish a trade white paper in Autumn, ahead of a Trade Bill.
A Department for International Trade spokesman said: “We have been very clear that we want a trade policy that is inclusive and transparent and which represents the whole of the United Kingdom.
“We will not be giving a running commentary on possible future trade policy”.
Plaid Cymru said any move to freeze out Wales would be “disgraceful”.
Welsh treasury spokesman, Jonathan Edwards MP, said: “If the UK leaves the customs union enabling it to strike trade deals, it is vital that no trade deal is signed without the endorsement of the Welsh Government.
“Otherwise the British government could expose key Welsh economic sectors and our public services, effectively supplanting the devolved settlement.
“Within the customs union, member states and sub national governments, like Wallonia in Belgium, can veto trade deals.
“It would be disgraceful if in post-Brexit UK, national governments within the British state are not able to defend their economic interests from Westminster politicians.
“The Trade Secretary would do well to remember that people in Wales have voted twice in binding referendums to empower our National Assembly.”