The Toffees have seen their £15m bid turned down by the Italian side, who remain determined to keep the midfielder at the club until the end of the season.
Everton's bid is thought to be way below Atalanta's valuation of the player, as they are demanding £30m-£34m for the 20-year-old.
Chelsea remain favourites to sign the Atalanta man, but are more likely to make a move in the summer.
Everton are hoping for an active January transfer window and have already bought Charlton's Ademola Lookman for around £10m and have already agreed a fee with Manchester United for Morgan Schneiderlin.