BERLIN—Chancellor Angela Merkel has offered government support to efforts to open up Germany to U.S. gas, a key concession to President Trump as he tries to loosen Russia’s grip on Europe’s largest energy market.

Over breakfast this month, the chancellor told a small group of lawmakers her government had decided to co-finance the construction of a €500 million ($576 million) liquefied natural gas shipping terminal in northern Germany, according to people familiar with the meeting, giving a crucial nudge to a project that had...