BRUSSELS—Western military commanders are concerned that large-scale Russian military exercises near the Baltic states in September pose heightened risks for a miscalculation that could lead to a crisis, allied officials said.

In a move likely to further heighten tensions, the U.S. Senate on Tuesday approved Montenegro’s bid to join the North Atlantic Treaty Organization, an initiative Moscow has bitterly opposed. Officials of the Balkan nation have said Moscow backed a coup attempt there last year to derail the effort. Russia...