Both Kennedy and Krushchev had expressed interest, to hedge their bets with rival lunar programs.
I doubt either side would cooperate on technologies too close to weapons requirements. So no cooperation on engine design, and not necessarily any cooperation on guidance.
I doubt the Apollo CSM could be replaced without significant delays, and the early Soyuz 7K-OK had… worse quality control and safety problems than the Apollo Block I.
I think the most likely opportunities for cooperation would involve an early Salyut-style space station, effectively moving the Apollo Applications Program earlier in the process, and/or involve discussions of spacesuits, safety, and research. Since the Soviets were using air, they might have objected to pure oxygen and prevented the Apollo 1 disaster. Since the Americans were testing alternative parachutes with Gemini, they might have insisted on parachute standards and averted the Soyuz 1 disaster.