Brexit secretary suggests parliament will not vote down plan and says negotiations could be completed within two years

Parliament will not vote down the deal struck by Theresa May with other European leaders when MPs are given a say on the outcome of negotiations, David Davis has said.

The Brexit secretary suggested MPs would vote in favour of whatever the prime minister manages to achieve as the UK electorate has already opted for Brexit in the referendum.

In a round of television interviews, he also revealed that he thought all of the UK’s negotiations and trade deals with the EU could be completed within two years. After that, Brexit could be phased in over one or two years to cushion the blow for businesses, Davis suggested, putting a possible timeframe on the “implementation phase” for the first time.

“At the end of two years we will have our deals. What may take longer is implementation,” he said. “There are lots and lots of countries who are very keen to do deals with us.”

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Asked what would happen if MPs block May’s deal, he said: “They won’t vote it down. This negotiation will succeed.”

If parliament refused to approve her plan, May could have a choice of returning to the negotiating table, walking away from the need for a deal or calling a snap general election to get a mandate for her plan.

Davis dismissed the idea that parliament would block any EU deal after May made a speech setting out her fullest plan yet for how Brexit will happen. She ruled out being part of the single market in order to achieve controls on immigration and indicated the UK will seek bespoke arrangements on customs.

Speaking on BBC’s Radio 4 Today programme, Davis said an implementation period to phase in Brexit would “not be too long”, suggesting one or two years would be enough. He also indicated it could cover border controls, rules affecting businesses and customs arrangements.

He said May’s speech outlining Brexit had prompted a positive reaction from other European capitals and cited “very strong interest in deals” on trade from countries such as New Zealand, Australia, Canada and the US.

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While Downing Street claims to have received positive noises from other European leaders, there is a barrage of negative headlines in the continental press, including Die Welt’s picture of Theresa May with the slogan “Little Britain”.

Davis’s optimistic tone was echoed by Boris Johnson, the foreign secretary, who wrote in his Daily Telegraph column that other countries were “already queuing up” for trade deals.

“We will no longer be part of the common commercial policy, or bound by the common external tariff, and we will no longer have our trade policy run by the EU commission,” he said. “That means – crucially – that we will be able to do new free trade deals with countries around the world. They are already queuing up.

“Under EU rules, we are not formally allowed to negotiate these new treaties until we leave. But there is nothing to say that ideas cannot be pencilled in.”

In her speech on Tuesday, May warned European leaders that the UK is prepared to crash out of the EU if she cannot negotiate a reasonable exit deal, prompting key figures in Brussels to say that the country was on track for a hard Brexit.