BEIRUT, Lebanon — With the Syrian rebel enclave of eastern Aleppo shriveling, Russia said on Monday that it would start talks with the United States this week on a deal for holdout insurgents to leave, and that any who refused would be regarded as terrorists subject to deadly assault.

Foreign Minister Sergey V. Lavrov of Russia, who conveyed the announcement at a news conference in Moscow, also said Russia and the United States were close to an agreement.

Mr. Lavrov added that Secretary of State John Kerry had submitted a proposal for the routes and timing of the fighters’ departure after meeting with Mr. Lavrov in Rome over the weekend.

American officials acknowledged the possibility of talks with the Russians in Geneva this week but would not confirm the substance of Mr. Lavrov’s remarks, which appeared to have taken them by surprise.