The thermodynamic process, in which internal energy of the system remains constant is

For an ideal gas, the internal energy depends only on temperature: $$U = nC_vT$$.

In an isothermal process, the temperature remains constant, and therefore the internal energy remains constant ($$\Delta U = 0$$).

By the first law of thermodynamics: $$\Delta Q = \Delta U + \Delta W = 0 + \Delta W = \Delta W$$

So in an isothermal process, all the heat supplied to the system is used to do work, while the internal energy stays constant.

Note: In an isochoric process, volume is constant ($$W = 0$$, but $$\Delta U$$ can change). In an adiabatic process, $$Q = 0$$ but $$U$$ changes. In an isobaric process, pressure is constant but $$U$$ can change.