Sage, 47, took over from Oliver Glasner in the summer but is already in danger of losing his job with Crystal Palace sat down in 18th.
The South London club have lost six of their nine Premier League games of the campaign so far and rumours regarding potential replacements are starting to circulate.
According to The Mirror, Barry, 40, Thomas Tuchel’s assistant at England, is one of them, should the Frenchman be axed.
Barry, who has also been Tuchel’s assistant at Chelsea and Bayern Munich, has never been a head coach, but is highly regarded as a tactician.
Like Tuchel, Barry signed a new deal with the Three Lions until the Euros in 2028.
