Current reference circuits are widely used in analog integrated circuit design. However, due to PVT and aging variations, the reference currents are mostly affected which requires reprogramming the reference circuit. Recently, memristors are investigated in many analog applications due to the programmability and non-volatility. In this paper, we introduce a simple programmable memristor-based circuit that can be used in current reference generators. The circuit is based on one memristor and one CMOS transistor to tune the resistance of the memristor. VTEAM memristor model is used to study the proposed circuit programmability. The necessary tuning conditions for the circuit are discussed. Then, the proposed circuit have been used in the well-known Beta-Multiplier current reference to generate a programmable current reference. The designed circuit has a reasonable current tuning range due to the limited range of the used memristor model. Different circuit simulations are provided using ST 65nm CMOS Technology.