"Call me if there’s anything I can do."

You mean it and are sincere but,Generally a person who is ill or needs help will never call. Whetherit is pride, embarrassment, or just that they don't want toinconvenience you, it most likely won't happen.So the person is need never calls,and you don’t know a graceful way to volunteer.They never get the help they need.You never have the chance to express your friendship.
Caregivers say instead,"May I bring dinner tomorrow?"I am available all day tomorrow, may I cook or drive or shop?Be specific about when you are available and what you can offer.It is a win win