The Lambert W function is used to study a linear consensus model in multi-agent systems with delay. In particular, the case where all nonzero eigenvalues of the Laplacian matrix are real is considered. An explicit expression is obtained for the delay ensuring the maximum degree of convergence. A formula for the maximum degree of convergence is derived. As proved, the maximum degree of convergence depends only on the maximum and minimum nonzero eigenvalues, while the other eigenvalues have no influence on this characteristic. The results presented are a basis for one still unsolved problem, i.e., the direct estimation of the convergence rate in multi-agent systems with a directed structure.