Yes, there is usually an implicitly applied electric field in the wire itself that gives rise to the current. Usually this electric field is ignored for the purposes of a simple static problem. However, if we have an alternating current, then this current is excited by an electromagnetic wave that propagates between the wire and the return/ground. This electromagnetic wave permeates the space around the wire. By convention, we do not regard the fields to penetrate the wire. A perfect conductor restricts all currents to the surface of the wire and prevents any fields from being present inside the wire. A realistic conductor can still be reasonably estimated to have this property as well.
author = "Assis, A. K. T. and Rodrigues, W. A. and Mania, A. J.",
title = "The electric field outside a stationary resistive wire carrying a constant current",
journal = "Foundations of Physics",
year = 1999,
volume = 29,
number = 5,
pages = "729-753",