Swansea are bottom of the Premier League following Saturday's 3-1 defeat to Everton and find themselves four points adrift of safety after losing nine of their previous 11 games.
Asked what has gone wrong at the club, Neville - sat next to Giggs - joked: "They didn't appoint him".
Frank De Boer became the bookmakers' favourite yesterday and is prominent on the shortlist but the club have also spoken to Aitor Karanka, the former Middlesbrough manager, and will continue to assess their options over the next few days.
"But we felt we couldn't leave it any longer and needed to make a change to give us the best chance of an uplift and a turnaround in fortunes with the club bottom of the Premier League".
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"Paul has been at the club for 12 months and what he achieved in the second half of last season to keep us in the top flight was a tremendous feat".
Clement was sacked by Swansea on Wednesday after a tough start to the season.
Clement has previously served as assistant manager at clubs including Bayern Munich, PSG, Real Madrid and Chelsea and it is perhaps just a matter of time before he is back in the game.
Swansea lost last season's leading scorer Fernando Llorente and midfielder Gylfi Sigurdsson during the summer transfer window, severely weakening their squad.