Boro are seeking a new manager with Rob Edwards now leaving for Wolves.
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The Sun says Liverpool legend Gerrard is a candidate for the job, along with fellow free agent Gary O'Neil and Valencia coach Carlos Corberan.
Gerrard turned down a return to Rangers after holding talks with directors last month. He remains in the Middle East after leaving Al-Ettifaq, his last management job.
Boro are happy to take their time during the international break, with the appointment unlikely to be made until early next week.