The simultaneous simulation of frictional contact mechanics and fluid flow in fractured geological media is a tightly coupled physical processes and a key component in the design of sustainable technologies for several subsurface appliactions, such as geothermal energy production, CO2 sequestration and underground gas storage. Typically, the aperture and slippage between the contact surfaces drive the fluid flow in the fractures, while the pressure variation perturbs the stress state in the surrounding medium and influences the contact mechanics itself. Despite several approaches have been advanced in recent years, including Embedded Discrete Fracture Models (EDFM) and Extended Finite Element Methods (XFEM), the coupled simulation of contact mechanics in hydraulically active fractures still poses several numerical issues, related to the stability of the discretization, the management of the non-linearity and the solution of the discretized problem. In this work, we focus on a blended finite element/finite volume method, where the porous medium is discretized by low-order continuous finite elements with nodal unknowns, cell-centered Lagrange multipliers are used to prescribe the contact constraints, and the fluid flow in the fractures is described by a classical two-point flux approximation scheme. This formulation is consistent, but is not uniformly inf-sup bounded and requires a stabilization. For the resulting 3x3 block Jacobian matrix, robust and efficient solution methods are not available, so we design a class of scalable preconditioning strategies based on the physically-informed block partitioning of the unknowns and state-of-the-art multigrid techniques. A set of numerical results concerning hydraulic fracturing applications illustrate the robustness of the proposed approach, its algorithmic scalability, and the computational performance on large-size realistic problems. The objective of the analysis is to identify the most promising numerical approach for an upcoming implementation on high performance computing machines.