The 23-year-old Chilean international, who can operate as either a defender, an attacking wing-back, or in midfield, has attracted attention from a host of clubs across Europe after a series of impressive displays in Serie A this season.
Udinese are keen to keep hold of one of their most prized possessions, particularly after losing former star player Alexis Sanchez to Barcelona in the summer.
However, reports suggest that the Zebrette will consider offers of around £10million for the South American star.
Several big clubs in Italy are also believed to be monitoring Isla's situation, but speculation in his homeland suggests that Liverpool are in fact leading the race for his signature.
Dalglish predicted a quiet January transfer window at Anfield, for the first team at least, despite constant speculation surrounding the Reds and a string of players based in both the UK, Europe and beyond.