Georgia manager Willy Sagnol is being linked to the post, as he may be free after Euro 2024, says The Sun.
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O'Shea took over on a temporary basis in February after Stephen Kenny's departure.
He managed the team for a 0-0 against Belgium and a 1-0 loss to Switzerland.
Ireland are not in Euro 2024, as they failed to qualify under manager Kenny in 2023.