There is a potential difference, but ions can not go through wires, right? Though there is a electric field, but there is no electron source, I am thinking the answer is no, or will there be some chemical reaction?
Let's start with the basics. The inside of the cell contains predominantly positive potassium ions, and negative phosphate ions, and other negative ions (e.g. from amino acids). The outside of the cell contains predominantly positive sodium ions, and negative chloride ions.
The cell however sets up a resting membrane potential, due to the cell's semi-permeable membrane, allowing predominantly only potassium to pass through the membrane. This causes potassium to flow along its concentration gradient towards the outside of the cell, until it is balanced by the electric potential that is consequently set up.
So we now have POSITIVE potassium ions on the outside of the cell and NEGATIVE phosphate ions (or other negative ions) on the inside of the cell, which aren't balanced. Now as the wire is placed between the inside and the outside of the cell, electrons from the negative phosphate ions will travel through the wire, towards the positive potassium ions on the outside of the cell.
Once this occurs, the potassium from the cell's interior can now continue to flow against its concentration gradient, because the previous potassium ions in the exterior no longer contribute to an opposing electric potential.
Additional note: As the K+ ion in the extra-cellular fluid gains electrons from the wire to form K, it will react almost immediately with water to form KOH and H2 (hydrogen gas). The K and OH will be soluble in the extra-cellular fluid, forming K+ and OH- ions. Over time the K+ ions will be "pumped" back into the cell's cytoplasm. (The left over OH- ions will temporarily increase the flow of K+ down its concentration gradient, etc.)
Just to clarify your ion/electron question - a neutral atom or molecule becomes ionized when it either loses one or more electrons, becoming positively charged, or gains electron(s), becoming negatively charged. Since the negatively charged phosphate (PO42-) ions have a surplus of electrons, they (the electrons) are free to flow "up" the wire and to the outside of the cell.
For electrons to flow trough the wire you'll need that the wire can have a redox reaction with the required solution, a simple voltage difference is not enough. Given that you may use a silver chloride wire or a platinum wire which can have these kind of reactions with the ions inside of the cell. If there`s no redox reaction, you can still see a transient capacitive current going trough the wire.