Reversibiliy and equilibrium
When a system is in thermodynamic equilibrium, there is no tendency for spontaneous change. In other words everything stays the same, the composition, temperature, pressure do not change over time, etc.
Therefore:
Tsurr is constant
and
Tsys is constant.
If you make an infinitesimal change, the system will tend to revert to the equilibrium.
So there is no issue.
When you are talking about a reversible process in thermodynamic terms, you're talking about a system which can be taken from its initial state to another state, and then back to the initial state without any net change to either the system or the surroundings.
The idea being that you make an infinitesimal change to some aspect of the surroundings, the system changes to a new state which is again in equilibrium with the surroundings, lather, rinse, repeat. Of course it would take infinite time for the process to be completed.