Suppose when we connect one wire with current conducting wire then it also becomes negatively charge which has potential to accept electrons. But when we break the circuit, the wire becomes neutral again and doesn't have potential to conduct current. How at one time it is negatively charge but when we break the circuit then suddenly it becomes neutral ?
Actually, it does retain a charge. But that charge is tiny, and it drains away as soon as you connect a voltmeter to measure it. You need to use an instrument like an electroscope, which has essentially infinite input impedance.
The wire remains charged. The wire has capacitance, but this is very small.
If you make the wire long enough or the voltage high enough, you will be able to detect it.
You've asked a question better suited for the physics forum, but I think the general answer to your question is that the wire never has an unbalanced charge. The electrons in it are being subjected to an electric field, causing them to move. The charge imbalance is in the power supply.