WASHINGTON—They had never met, and they had had only harsh words toward each other during the campaign. But President Barack Obama and his successor, Donald Trump, are discussing policy and politics with more frequency than many anticipated after this past year’s bitterly fought election.

The two men have spoken a handful of times by phone since their 90-minute meeting in the Oval Office last month, setting aside resentments that built up during the campaign, advisers to both said. Though it is hardly a “bromance,” both men...