For an autonomous linear system of differential equations with delay, we find asymptotic formulas for the analytic dependence of regularized solutions of this system on a regularization parameter on a finite closed interval of the negative semiaxis. We use the Tychonoff regularization method with a stabilizing functional not generating a compact set in the state space. The problem is solved for sufficiently smooth initial functions, which, however, do not satisfy the conditions that guarantee a continuous extension of solutions to decreasing time.