Adkins is feeling the heat after seeing his Saints sink into the Premier League relegation zone with just one win all season.
Cork said: "To be fair, the gaffer was having to face speculation about his future after just a few weeks of the season because results hadn't been that great.
"But to suggest things like that is just silly in my opinion. He's done a fantastic, remarkable job at Southampton and just doesn't become a bad manager in such a short space of time.
"It's a shame. He's done so well here and the fixtures haven't been kind either. It's far too soon to judge him because the Premier League is so much harder than the Championship."