The Mirror says the Serie A side have had a shocking start to the campaign with most of the blame aimed at their back four.
Juve coach Luigi Del Neri is now determined to strengthen his defence in the January window and he is keen on Shawcross.
The 6ft 3ins former Manchester United trainee's no-nonsense defending is deemed perfect for the Italian league.
Shawcross, 22, has become one of England's best young central defenders and he has featured in two of Fabio Capello's squads.
But Stoke have made it clear they will not sell their prized asset.