LONDON — Ridiculed by the right-wing tabloid media and ignored by Prime Minister Theresa May as she pursued plans for a clean break with the European Union, Britain’s pro-Europeans suddenly have something they have long wanted: leverage.

After the recent stunning general election, in which her governing Conservative Party lost its parliamentary majority, Mrs. May faces pressure from both inside and outside the party to soften her plans to exit the bloc, a process known as Brexit, as talks are set to begin on Monday.

The pro-Europe Britons’ demands that Mrs. May maintain closer ties to the European Union have grown louder and more assertive — in particular the calls to keep Britain in Europe’s customs union, which provides tariff-free access to Continental markets and helps integrate the British and European economies.

For the first time since the referendum on Britain’s exit, there is “an opportunity to have a much better relationship with the European Union,” said Roland Rudd, a senior figure in the defeated “remain” campaign and founder of Finsbury, a communications company.