George Osborne has warned that leaving the European Union without any trade deal would amount to "the biggest act of protectionism in British history."
The ex-chancellor told business leaders that the UK could not rely on new trade deals beyond Europe after Brexit.
But Boris Johnson said the UK could be "ever-more European and ever-more internationalist" at the same time.
The foreign secretary said there was "every reason" to believe a "win-win" EU deal could be done within two years.
Downing Street said Prime Minister Theresa May was "confident and optimistic" about striking a comprehensive free trade agreement.
No 10 also said Brexit Secretary David Davis had told cabinet on Tuesday that the UK must prepare itself for the "unlikely scenario" this could not be achieved.
In another development, a senior Nissan executive has told MPs the car maker may "adjust" its business in the UK dependent on the outcome of Brexit.
Colin Lawther said Nissan, which in October committed to building two new models at its Sunderland plant, would "constantly review" its decision in light of any major changes and the firm believed remaining in the customs union was vital.
The government is expected to notify the EU of its intention to leave next month, paving the way for a two-year negotiation over the terms of exit and the shape of the UK's future relationship with the EU.
Theresa May has said no deal would be better than a bad one and indicated she would be prepared to see the UK revert to World Trade Organisation trading rules in the absence of any agreement.
Mr Osborne, who campaigned to remain in the EU but was sacked by Mrs May after she took power in July, told business leaders that he recognised that the UK would be leaving the EU and he was not seeking to "fight the last war".
But he said leaving the EU without a proper agreement would impose significant costs on the British economy, with up to half of all British exports facing tariffs and additional regulations for trading with EU.
"Let's make sure that we go on doing trade with our biggest export market," he said.
"Otherwise withdrawing from the single market will be the biggest single act of protectionism in the history of the United Kingdom.
"And no amount of trade deals with New Zealand are going to replace the trade that we do at the moment with our big European neighbours."
He also expressed doubts about whether a special customs arrangement could be agreed with the EU replicating the existing benefits of the customs union.
Critics of Brexit, including former prime ministers Tony Blair and Sir John Major, have warned that there is little chance any new trade deal could match the advantages of the UK's current membership of the single market.
But speaking at the same conference, Mr Johnson said he had had enough of hearing people "droning on and moaning about the state of the world".
"Since June 2016, the sky has obstinately failed to fall in," he said.
The UK's interests, he added, had not been best served by delegating its trade policy to the EU and that the UK had a "unique global quality" which meant it would be able to maximise the opportunities provided by Brexit.
He said he regretted the fact that globalisation had become a "boo word in the political lexicon", saying it had been a "liberating and enriching force" in lifting millions out of poverty and contributing to peace and prosperity in Europe.
The UK, he added, had an "unrivalled opportunity" to do a "fantastic" deal with the EU, given their trading standards, rules and tariffs were already harmonised.
"Of course we remain committed to our European markets, perhaps more so than ever but we need to think globally again because a global Britain is a safer, more successful and more prosperous Britain."
Ministers have said that although no new trade deals can be signed until the UK leaves the EU, a host of countries are keen to talk to the UK about the prospects.