The 20-year-old has been earning just £850 per week on an academy contract at St. James' Park.
Toon had offered a contract worth £15,000 per week, but Longstaff turned it down.
Sky Sports says Udinese's five-year contract is worth double that amount, along with a significant signing-on fee.
Losing homegrown Longstaff would be a travesty for the Magpies.
The midfielder is desperate to stay at his boyhood club, but offer from Italy is tempting him to leave.