  Autonomous entanglement engines have recently been proposed to generate
steady-state bipartite and multipartite entanglement exploiting only incoherent
interactions with thermal baths at different temperatures. In this work, we
investigate the interplay between heat current and entanglement in a two-qubit
entanglement engine, deriving a critical heat current for successful operation
of the engine, i.e. a cut-off above which entanglement is present. The heat
current can thus be seen as a witness to the presence of entanglement. In the
regime of weak-inter qubit coupling, we also investigate the effect of two
experimentally relevant parameters for the qubits, the energy detuning and
tunnelling, on the entanglement production. Finally, we show that the regime of
strong inter-qubit coupling provides no clear advantage over the weak regime,
in the context of out-of-equilibrium entanglement engines.
