The melting of ice is a reversible process according to most sources I have seen. However, the melting of ice increases the total entropy of the universe which shouldn't be possible if the process was reversible.
Can someone help clear up this concept?
The melting of ice, in the idealized case, is reversible. Here, we have a mass of ice at 0°C (at standard conditions) and an infinitely large heat reservoir (or environment idealized to have infinite heat capacity) also at 0°C. Thermal energy $Q$ flows from the reservoir to the ice. (Don't ask why. When we consider reversible idealizations in thermodynamics, there's often no reason for anything to occur.) The melting of the ice produces a notable entropy increase because of the increased mobility of the water molecules. The same entropy (Q/273 K) is removed from the heat reservoir. Entropy is transferred, and none is generated.
Melting of ice when you personally observe it is irreversible when you expand the scope to include the cause. Ultimately, something hotter than 0°C (at 1°C, say) heats the ice to melt it. The flow of energy down a temperature gradient generates entropy. (The surrounding environment loses entropy ~Q/274 K, and the ice gains entropy Q/273 K, which is a larger number.) Correspondingly, the process is thermodynamically spontaneous, as is everything you actually observe. Moreover, the entire real, observable process is irreversible. Does this answer your question?
Why do you think the entropy increases? The change in entropy of the heat source is $−/_{\rm melt}$ and the change in entropy of the water/ice mix is $+/_{\rm melt}$ with the same $$, so the sum of the entropy changes is zero.
The water in contact with the ice is always at the melting temperture, so no entropy is generated by the melting itself --- but if the ice is in warm water then entropy is generated by the heat conduction from the warm water to the melting-point water.
