LONDON—The U.K.’s governing Conservative Party opened the way to some financial settlement with the European Union after Brexit, appearing to soften its stance on a likely early sticking point in coming Brexit negotiations, as it published its manifesto for a national election next month.

The party, which opinion polls suggest will win the June 8 election with an increased majority, confirmed it intended to extract Britain from the bloc’s customs union as well as its single market for goods and services.

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