Morosini, 25, was on loan at Livorno when he collapsed and died during a clash with Pescara on Saturday.
Udinese has confirmed plans to look after the player's only remaining family member, his severely disabled sister.
The player's sister relied on Morosini for financial support following the death of their parents, and Di Natale has assured that Udinese will now look after her.
"We know the situation of his sister and we as a team, the club, and Udinese for Life have decided to help her because she is in real need," Di Natale said.
"It is essential to stay by the side of Piermario’s sister for her entire life. She needs us and we want to help, both for her and for Mario.
"When I saw the scene on the TV it was immediately clear that it was something serious. It was a very bad moment. He was an exceptional lad, so full of life. Despite all the problems he had, he was always at the disposal of the team and every day he gave us all strength."
The Italian Football Association (FIGC) opted to cancel all Serie A fixtures in the wake of Morosini's tragic death.
