The Internet is very important for bitcoin, as outside of it bitcoin simply can not function. Therefore, for all operations with cryptocurrency you need access to the network. But perhaps in the future this problem will be solved, as it can really create obstacles to the implementation of bitcoin as a currency.
They are both really connected with each other, so it's impossible to separate them to make transactions or offline transaction. But if for receive a bitcoin, it's not mandatory to connected through internet, but the increasing balance isn't visible.