That source, and another in the Conservative party, said May's misjudged election gamble had undermined her authority, leaving her in the thrall of the two wings of her party that have differing views for Brexit - "purists" who want a clean break and "remainers" pressing for close ties.
The Government is being urged to put the economy first and secure a transitional Brexit deal with the EU to ensure tariff-free access to European markets.
Unresolved is everything from the status of European Union citizens living in Britain, to intelligence sharing, to the future of tens of thousands of British jobs that could be wiped out if businesses move to Europe to avoid new trade barriers.
The vote came as a profound shock to Brussels against a backdrop of rising anti-EU sentiment, with many - including now US President Donald Trump - predicting the bloc's eventual break-up.
May has also threatened to walk away from negotiations with "no deal". British ministers insist the final sum will not be anything like that.
"I think it's premature to speculate about the outcome on the first day of talks", she told reporters in Berlin.
"Positive growth numbers on the continent are coming into focus now some of the contentious national elections are out of the way".
Theresa May is pressing for a relationship with the European Union which no other nation outside the bloc has achieved.
Unemployment is at its lowest level in more than 40 years, and many companies in hospitality, healthcare, tech and construction are struggling to find staff.
The dialogue will be managed by Mr Barnier's deputy Sabine Weyand, and Olly Robbins, the permanent secretary at the department for exiting the EU.
European Union member states pay into a communal budget, which finances infrastructure projects, social programs, scientific research, farm subsidies and pensions for European Union bureaucrats.
Mr Barnier did not put a figure on the settlement likely to be required from the United Kingdom, estimated by some in Brussels at as much as 100 billion euro (£88bn).
Britain has balked at figures in that range.
She will also go into those negotiations lacking the strong mandate that she had sought in those elections.
He said: "We have to commit ourselves now mutually to guarantee rights to citizens on either side of the Channel so they can continue their lives as in the past".
After the first day of the crucial negotiations which have the potential to shape the UK's economic and political future for a generation, it was agreed that working groups of officials would aim to make progress on the issues of citizens' rights, the UK's financial settlement - the so-called divorce bill - and other issues to do with separation.
Today's talks are expected to be the first of four or five sets of talks to be held over the summer concerning the first phase of the negotiations, namely the terms of Britain's departure from the EU.
May has yet to change tack formally, and doing so would risk infuriating euroskeptics in her Tory party who might still prove strong enough to scupper a deal or to topple her as leader if they believe she's backsliding. British treasury chief Philip Hammond last week said that Britain should prioritize business interests as it negotiates.
He suggested the Tories might get May to concede on parts of the Brexit divorce that are important to the European Union but unpopular for UK Brexiteers, allow her take the flak and then drop her for a new Prime Minister who would have a free run at negotiating the trade deal. Both sides want transparency to be the default.
Another important point for the British Government is the border between the two Irish, as London wants some formula that avoids to return to the border control and to prevent to harm the process of peace sealed with the Agreement of Good Friday of 1998.
- Ivana Kottasova contributed reporting.