The International Trade Secretary has previously said he would be happy with interim measures that lasted a “few months” but now believes it should be wrapped up by the next general election.
It comes after Environment Secretary Michael Gove said the Cabinet was “united” around a transitional Brexit deal to allow continued access to migrant labour and provide economic stability.
Mr Fox said the time it takes Britain to quit was “not a huge deal” but he believed people did not want it “dragging on”.
He told BBC One’s The Andrew Marr Show: “Having waited for over 40 years to leave the European Union, 24 months would be a rounding error.
“It’s not a huge deal and neither is it an ideological one.
“I think we would want to get it out of the way before the election, I don’t think people would want to have it dragging on.”
But the Federation of German Industries, the BDI, warned that business would be hit without a long transitional deal.
Klaus Deutsch told BBC Radio 4’s The World This Weekend: “We would favour a comprehensive agreement. But the most important thing is legal certainty in the period from A to B. If you don’t have a transition period of many years then there will be a huge disruption to all sorts of businesses.
“The concern of business is unless you get a clear-cut and legally safe agreement you can’t sell pharmaceuticals, or cars or what have you, across the channel. You have to stop business, divest, change business models.”
Mr Deutsch said it was “completely unlikely” that German Chancellor Angela Merkel would soften her stance.
“The importance of the European Union for German corporates is even higher than the importance of a bilateral relationship with the United Kingdom.
“So the priority of safeguarding the four freedoms and the unity of the European Union is much more important than one economic relationship. There are a lot of illusions, it won’t happen.”
Jeremy Corbyn, meanwhile, has been accused of parroting lies after he insisted the UK must quit the single market because it is leaving the European Union.
The Labour leader said he still did not have a clear position on whether Britain should remain part of the customs union but the UK must leave the single market as it is “inextricably” linked with EU membership.
Labour former frontbencher Chuka Umunna, who led a rebellion against the party leadership over its position on the single market, pointed to countries outside the bloc that are part of the trading area, and warned that most Labour members want Britain to remain in the market.
Mr Corbyn said Labour had made it clear it wanted tariff-free trade with the EU but had not “jumped on either side” of the fence on customs union membership.
He told BBC One’s The Andrew Marr Show: “The single market is dependent on membership of the EU. What we have said all along is that we want a tariff-free trade access to the European market and a partnership with Europe in the future.
“The two things are inextricably linked, so the question then is the kind of trade relationship of the future and we have made it very clear we want a tariff- free trade access with the European market.
“We haven’t jumped on either side of that fence but, again, the customs union is part of the European Union.”
Mr Umunna pointed out that non-EU countries Iceland, Lichtenstein and Norway are part of the single market.
“The overwhelming majority of Labour members think we should be fighting to stay in the single market – let’s do it,” he said.
Meanwhile, new Liberal Democrat leader Sir Vince Cable said he had been in talks with members of the shadow cabinet and Conservative backbenchers over the fight against Brexit, and said that Labour MPs were being “intimidated” and told to “toe the line or else”.