The Sun states that Barry, who is a top-rated young English coach, was wanted by the Terriers, but only if he committed immediately.
The source adds that Barry has turned down the chance to manage Huddersfield.
He has a commitment with Roberto Martinez and Belgium for the 2022 World Cup, where he will assist Martinez.
There have been other Football League clubs interested in Barry as well.
However, his commitment with Belgium is iron clad, which means any future jobs will have to wait until January.