The UK wants to start talking about its post-Brexit relationship with the EU in order to reassure anxious businesses, citizens and investors, but the union has insisted that first exit arrangements should be determined.
"We are both clear that during this period the United Kingdom will be outside the customs union and will be a "third country", not a party to EU treaties".
Hammond and Fox are seen as the two ministers at opposite ends of the soft-hard Brexit spectrum and their joint article, which also said the transitional period would be "time-limited", was meant to quash speculation that the cabinet is divided on the issue.
"By triggering Article 50 in March of this year we leave in March 2019, what the transition period will look like is down to the negotiations", he said during a surprise appearance at Alex Salmond's show at the Edinburgh Festival Fringe.
"The DUP has been discussing these matters with the government for some time and we support the concept of maintaining a common travel area between the United Kingdom and Irish Republic and share the government's view that modern technology can help with monitoring the movement of goods across the border in the event of the UK reaching a new free trade and customs agreement with the European Union", he said.
"We've been crystal clear that issues around our withdrawal and our future partnership are inextricably linked, and the negotiations so far have reinforced that view".
Prime Minister Theresa May will hope the intervention of the two ministers will cool temperatures in the Tory ranks amid divisions over Brexit and speculation of a possible leadership challenge when MPs return to Westminster in September.
The bloc's chief negotiator, Michel Barnier, said last month there was "a clock ticking" on the talks.
Since May lost her parliamentary majority in a election gamble in June, infighting between members of her cabinet has appeared, with disagreements on issues including whether freedom of movement of European Union nationals should continue after Britain leaves the bloc in 2019.
The Sunday Telegraph reports one proposal will involve offering Irish citizens free movement in and out of the UK.
However، sources at London's Department for Exiting the European Union insisted Sunday that the two were "inextricably linked".
"Nobody voted to be poorer previous year but that is exactly what will happen if the Government continues to put Eurosceptic dogma ahead of the national interest".
The UK formally triggered the Brexit process on March 29 and divorce negotiations officially began on June 19.