Swans have sacked Alan Sheehan this week after a disappointing Championship season so far.
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They're now seeking a new manager and the Daily Mail says Manchester United Champions League winner Carrick is under consideration.
The Geordie left Boro at the end of last season and has remained out of the game.
Former Swansea manager Brendan Rodgers, who resigned from Celtic last month, is also being discussed inside the Swans board room.