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, for a two-qubit entanglement engine in the steady-state regime, we derive an analytical relation between the heat flow and entanglement by showing there exists 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 between the qubits. We further show that while energy detuning and tunnelling in general reduce the amount of entanglement, both effects can compensate each other for specific values.