20-year-old Longstaff turned down a £15,000-a-week offer from Newcastle earlier in the season, and has received no improved proposal.
Udinese are ready to double that offer and also hand Longstaff a big signing-on fee.
Due to his age, Udinese would have to pay just £400,000 in compensation to Newcastle.
Longstaff wants to stay at Newcastle, but the bumper offer from Italy leaves him with a big decision to make.
The young midfielder burst on to the scene in October when he scored the winner in a 1-0 win against Manchester United.
Inter and Marseille have been linked with moves for Longstaff, but only Udinese have made a concrete offer.
Longstaff’s older brother Sean, 22, could also quit St. James’ Park with his contract expiring in 18 months and clubs interested.