The Log Cabin Republicans, a conservative gay and lesbian political organization, declined on Saturday to endorse Donald J. Trump for president, revealing split feelings within the group over whether the Republican nominee is really the advocate of gay rights he claims to be.

Citing the Russian proverb that Ronald Reagan used to sum up his feelings on negotiating with the Soviets — “Trust but verify” — the group said in a written statement that it was not yet prepared to fully support Mr. Trump’s candidacy.

“Should Mr. Trump become our nation’s next president, Log Cabin Republicans welcomes the opportunity to work with his administration,” the statement said. “Until and unless that happens, our trust would be misplaced.”