HELSINKI—President Donald Trump enters his summit with Russian President Vladimir Putin with a determined effort to lower expectations, as his top aides say the goal is to open a conversation rather than accomplish a reset of U.S.-Russian relations.

U.S. lawmakers urged Mr. Trump to press the case against Mr. Putin for meddling in the 2016 elections. But while the Monday meeting will mark a symbolic end to the American effort to isolate Russia after its 2014 annexation of Crimea, the question persists: What will the U.S. get...