WASHINGTON—The Obama administration threatened to pull out of talks with Russia over a collapsed cease-fire in Syria and has renewed an internal debate over giving rebels more firepower to fend off a stepped-up Russian and Syrian assault on their Aleppo stronghold, U.S. officials said.

The White House put the weaponry debate on hold earlier this year to give Secretary of State John Kerry an opportunity to try to negotiate a cease-fire with his counterpart in Russia.

Mr....