  In this paper, the realization-free model approximation problem, as stated in
\cite{mayo2007framework,beattie2012realization}, is revisited in the case where
the interpolating model might be time-delay dependent. To this aim, the Loewner
framework, initially settled for delay-free realization, is firstly generalized
to the single delay case. Secondly, the (infinite) model approximation
$\mathcal{H}_2$ optimality conditions are established through the use of the
Lambert functions. Finally, a numerically effective iterative scheme, named
\textbf{dTF-IRKA}, similar to the \textbf{TF-IRKA}
\cite{beattie2012realization}, is proposed to reach a part of the
aforementioned optimality conditions. The proposed method validity and interest
are assessed on different numerical examples.
