CAIRO — An attempt by President Trump to break the stalemate that has divided the wealthiest countries in the Middle East ended in failure on Saturday, when leaders from Qatar and Saudi Arabia, after speaking by phone for the first time in months, exchanged dueling, contradictory statements.

Mr. Trump arranged the call, which took place late on Friday, and promised a breakthrough in the bitter dispute that has plunged the Persian Gulf into turmoil and has threatened American security interests.

Since June, Saudi Arabia has led the United Arab Emirates, Egypt and Bahrain in imposing a punishing trade and transport boycott against tiny, gas-rich Qatar, accusing it of financing terrorism and having overly cozy relations with Iran. Qatar has rejected the charges, countering that its rivals are seeking to curb its sovereignty and tame its influential television channel Al Jazeera.

Mr. Trump stepped into the frame this past week, offering his services as a mediator and predicting a quick victory.