Saints reportedly had a bid turned down for Sturaro during January, when Ronald Koeman was still in charge at St Mary’s.
And, despite a change of manager this summer, the south coast club’s interest in the player does not appear to have waned, according to reports in Italy.
Despite Juventus being insistent that the 23-year-old is not for sale this summer, the Gazzetta dello Sport have claimed that Saints tabled the bid to test their resolve to keep him.
Sturaro started just 11 Serie A games and it has been suggested that Juve may decide to cash in on him if the money on offer is good enough.
However an initial bid of €15m, which is £12.8m at the current exchange rate, is not enough for Juve who have rejected the offer.
Lazio are also monitoring the situation after placing him on their wanted list this summer. Chelsea and Palace, meanwhile, were linked with him last season, and may also decide to re-enter the race.