I know that for a body to complete motion in a vertical circle, the initial velocity at the lowest point must be such that the tension at the highest point is zero, what if the velocity is greater than the said value and the tension at the highest point is not zero?
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the condition you describe is a minimal condition under which the movement will be a circular motion. If not the mass would not reach the top. If the forces in radial direction are greater the motion will continue to be circular the only difference would be angular speed and acceleration.
