Since the team's opening-day win over Burnley in the Premier League, three defeats and one draw have followed.
The Swans were also knocked out of the EFL Cup by Manchester City in the third round last night.
According to ESPN, the Swansea hierarchy have placed Giggs at the top of their wanted list in the event that the manager's position becomes vacant.
The former Wales international decided to go it alone in the world of management this summer when he turned down a role at United after it was made clear that Rui Faria would be Jose Mourinho's number two.
The report claims that Swansea chairman Huw Jenkins is an admirer of Giggs, who is currently looking for his first break in management.