In a real Hilbert space, let the VIP indicate a variational inequality problem with Lipschitzian, pseudomonotone operator, and let the FPP denote a fixed-point problem of a quasi-nonexpansive operator with a demiclosedness property. This article designs two quasi-inertial Tseng’s extragradient algorithms with adaptive stepsizes for finding a common solution of the VIP and FPP. The proposed algorithms are based on inertial Tseng’s extragradient approach with adaptive stepsizes, hybrid steepest-descent algorithm, and viscosity approximation technique. Under appropriate assumptions, it is proven that the sequences constructed by the proposed algorithms converge strongly to a common solution of the VIP and FPP. Finally, the main results are applied to deal with the VIP and FPP in an illustrating example.