In Brussels, the belief is that if Prime Minister Theresa May wins a large majority as expected on June 8, she will no longer have to worry so much about upsetting hardliners in her Conservative Party.
That would give her space to consider options that are anathema to core Brexiteers, such as a transitional deal until 2022 in which free movement and the jurisdiction of the European Court of Justice could continue.
European Parliament President Antonio Tajani said as much when he visited May in London on Thursday, adding that a new government before talks start later in June is "good not only for the UK but for us."
That view is widely shared. A European source close to the negotiations told AFP that a May victory would give a "a strong leader in London who could negotiate with us, with the strong support of her electorate".
The current mood is more promising than when May formally triggered the two-year Brexit process last month, a step that saw both sides taking tough positions on the talks ahead.
London and Brussels have been at odds on whether talks on the divorce arrangements, including costs, should take place at the same time as negotiations for a trade deal between post-Brexit Britain and the EU.
"She would be less at risk of being taken hostage by supporters of a 'hard Brexit'... and we will see what she really wants to do," said Ignacio Molina, of the Elcano Royal Institute think tank in Madrid.
He said May would have the "political capital" to take her own decisions if she increases her centre-right party's current slim majority.
The Conservatives won that in 2015 under David Cameron, who resigned after last June's shock Brexit referendum result. May is not only keen to win her own mandate but also to be less at the mercy of her party's anti-EU right-wing when it comes to parliamentary votes.
"The good news, from a European point of view, is that she would be less fragile for all the concessions that she is going to have to make," another European diplomatic source said.
Since her election announcement May has adopted a new and softer mantra on a possible Brexit deal, said Giles Merritt of Friends of Europe, another think tank.
"Instead (she will) be open to the sort of arrangements with the EU that hardliners in her Conservative Party have been condemning as a 'soft Brexit'," Merritt said in a commentary.
Simon Tilford of the Centre for European Reform predicted that, despite the divisive nature of the Brexit referendum, May would win a massive majority of 130-150 seats because of the opposition Labour party's weakness.
If so the "positive scenario is that she wants that (a majority) so she's not vulnerable to attacks from the eurosceptic right," Tilford told AFP.
"She has in recent weeks sounded much more flexible about freedom of movement, much more flexible about the European Court of Justice and she now seems to understand that a transition agreement between Britain and the EU will require Britain to abide by ECJ rulings and continue freedom of movement," he said.
May has so far insisted that Brexit means tougher immigration rules, including ending free movement of EU citizens, and of a complete break from any kind of power of the EU's Luxembourg-based top court.
But Britain should not hope for any favours from the other 27 EU countries just because May has a stronger mandate, Tilford said.
"I don't think May having a bigger parliamentary majority, having a mandate to pursue Brexit as she puts it, will make the EU side more accommodating or more flexible towards Britain," he said.
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