We study a linear extension of the Constantin–Lax–Majda (CLM) model, a prototypical nonlocal equationfor vortex stretching, and construct its integrable time discretization. By introducing the complex variablez=Hw+iw, the extended equation is reduced to a Riccati equation and solved in closed form. The additionallinear term shifts the fnite-time singularity while preserving exact solvability. Exploiting this linearizablestructure, we derive an exact discrete-time scheme by discretizing the Riccati equation rather than the originalnonlocal equation. The resulting system is an explicitly solvable discrete nonlocal evolution equation involvingthe Hilbert transform. Closed-form discrete solutions, a precise blow-up criterion, and the continuum limitare obtained. This provides an example of an integrable discrete nonlocal system with Hilbert-transforminteractions and extends CLM-type dynamics into the framework of integrable discrete equations.