Michael Gove has promised a "pragmatic" approach to Brexit in response to suggestions Britain could maintain free movement for EU citizens during a transition period following the official separation from Brussels.
The Environment Secretary said decisions on an "implementation period" would be made "in the best interests of our economy", while fellow Brexiteer Liam Fox said he was prepared to wait "another couple of years" for a full separation from Brussels.
Their comments indicated a widening of Cabinet support for a transitional deal between the UK leaving the EU in March 2019 and a new trading arrangement being introduced.
But the terms of any such deal are likely to be viewed with suspicion by hardline Eurosceptics.
A report in The Times said Prime Minister Theresa May is ready to offer free movement for two years under a plan drawn up by Chancellor Philip Hammond, while the Guardian quoted "a senior Cabinet source" as saying that the period could last for three or even four years.
It is thought that Mr Hammond believes he has won backing within the Cabinet for a transition to prevent disruption to business caused by a sudden "cliff-edge" move to new arrangements on March 29 2019, when Brexit is due to happen.
Answering questions following his first speech as Environment Secretary, Mr Gove would not be drawn on the details of any potential transitional deal - or implementation period as ministers refer to it - but said the Government wanted the process to be as smooth as possible.
Speaking in Woking, Surrey, he said: "I think when it comes to an implementation period it should be driven by a pragmatic judgment about what we need in the best interests of our economy and in the best interests of guaranteeing a smooth exit from the EU, in line with the result the British people voted for just over a year ago.
"Pragmatism is the watchword."
Pushed on whether freedom of movement would continue to apply, he said: "I think the judgment we need to make about future migration policy should and be shaped by what's in the interests of our economy, consistent with recognising the instruction the British people gave us last year."
International Trade Secretary Liam Fox has said a trade deal with the EU should be the easiest in history because both sides currently had the same rules, but was relaxed about the prospect of a transitional period.
"Having waited 43 years, another couple of years - if we can get it to work well for the country and for our European partners, and we can do it for the stability of British business and for our own prosperity - that seems to me just a common sense thing to do," he told Sky News.
Mrs May stressed her backing for an implementation period when speaking with business leaders at the first of a series of quarterly Downing Street forums on Brexit.
"The Prime Minister reiterated that the Government's overarching goal is for a smooth, orderly exit culminating in a comprehensive free trade deal with the EU, with a period of implementation in order to avoid any cliff edges," said a Downing Street spokesman.
But leading Brexit-backing Conservative MP Jacob Rees-Mogg said calls for a transition period were being used as a mask for attempts to overturn the result of last year's referendum.
"If we are subject to the rules of the single market and the regulations of the single market, and subject to the fiat of the European Court of Justice, we are paying for the privilege and we can't do free trade deals with the rest of the world, then we are in the EU," Mr Rees-Mogg told BBC2's Newsnight.
"In the old and tired phrase, if it looks like a duck, walks like a duck and quacks like a duck, it's a duck."
Fellow Leave-backing Conservative MP Anne-Marie Trevelyan said Britain would have to overcome its skills gap before cutting immigration.
The Berwick-upon-Tweed MP told BBC Radio 4's Today programme: "Numbers will start to reduce as we fix our skills gap and we work out with businesses what it is they need.
"Nobody wants to see a system where, if you can't find agricultural workers to pick your strawberries, you allow that business to fail. That is not the solution."
Ukip's interim leader Steve Crowther said: "Extending freedom of movement for two, three or four years produces no obvious benefit to anyone. It is an EU principle, so ending it in 2019 or 2023 makes no difference to them.
"Business wants certainty, and that comes from sticking to the timetable, negotiating robustly and introducing clear border control policies that enable us to access the skills we need.
"Since the election, Theresa May is badly holed and unseaworthy, and the Remainer Philip Hammond – who was on his way out of the door before June 8 – now sees an opportunity to fudge, delay and obfuscate until the end of the current parliament, to try and get the decision reversed."
Tory Eurosceptic Peter Bone said it would be "unacceptable" for free movement to continue under a transitional deal.
"If you are saying a transitional deal means free movement for another two, four, six years, that is absolutely unacceptable," he said.
On the terms of any transitional arrangements he told BBC Radio 4's World At One: "There's a complete difference between minor technical things that need to go on and major things like free movement.
"Free movement has to end no later than March 31 2019, and I think most Conservative MPs would say that, the country would say that and, absolutely the most important thing, I think Mrs May would say that."