The British Prime Minister had convened general elections to try to widen her majority and to negotiate with the European Union from a position of strength, but Mrs May lost her bet.
The announcement followed speculation that talks may have to be delayed because of the inconclusive outcome of last week's General Election, which has already forced the State Opening of Parliament to be put back from June 19 to 21.
Does the British government still want to leave the customs union and the single market?
An EU official close to the matter said the "softer Brexit" talk could be "productive" and help progress in the first months, where the British attitude to discussing the financial settlement "will be the first serious test of the negotiations".
"There is a unity of goal among people in the United Kingdom", May said following a meeting with French President Emmanuel Macron in Paris. The UK government meanwhile had only indicated that they wanted to push a hard line Brexit as they repeatedly emphasised "No deal is better than a bad deal'".
The report cited unnamed sources, and the finance ministry declined to comment.
Other EU governments will be happy to let Britain keep trade open as it would limit the hit to their own economies, officials told Reuters, though they are not ready to ease conditions that May would struggle to sell to her party's Brexit hardliners. They have struggled for years with Irish nationalists, who want the British province to join a united Ireland.
The Irish republican Sinn Fein party - which won seven seats in the election although their MPs traditionally do not take up their seats in protest - is also wary of the alliance.
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Ministers are likely to have their first political discussion on the issue.
"We would restore faith in politics if we could show that this parliament can at least function in presenting a view in the national interest which would command a majority on a cross-party basis", said veteran pro-European Conservative lawmaker Ken Clarke.
Former prime minister David Cameron said May needed to listen to opposition parties, and that there would be pressure for a softer Brexit that would give greater priority to a close trading deal with the EU.
May faces a hard balancing act.
He wouldn't be drawn on whether he supported Britain's continued membership of the single market or the customs union. But the outlook is strong for the pound, "considering the potential of the British economy". Government bond prices suffered heavy losses on Tuesday after consumer price inflation jumped to 2.9 percent in May.
Speaking alongside the President at the Elysee Palace, she said: "I confirmed to President Macron that the timetable for the Brexit negotiation remains on course and will begin next week".
The veteran conservative predicted that Britain would regret its departure from the bloc at some point in the future.
"My clear view - and I believe the view of the majority of people in Britain - is that we should prioritize protecting jobs, protecting economic growth and protecting prosperity as we enter those negotiations and take them forward", the U.K. Treasury chief said.