Not literally a gentle push, but very little rocket action is needed. The gravity well of an asteroid is tiny. The rest can be done with the correct slingshot maneuvers, the problem is calculating it. I am sure I have read something or other from NASA about it.
It's not the asteroid gravity that's the issue, it's the solar gravity field, you still have to perform an orbital transfer from the asteroid orbit to Earth orbit unless you want to leave the computer there and do batch jobs with significant latency.