Jurgen Klopp's side are interested in watching the development of the England Under-19 international, who spent the start of the season on loan to League Two side Crawley Town.
The 18-year-old, who was re-called in October to provide back-up to Rob Elliot after injuries to Tim Krul and Karl Darlow, has also attracted the interest of Southampton, Crystal Palace and Tottenham Hotspur.
Woodman sees his Newcastle contract expire at the end of the season and is still yet to agree a new deal, which has put the Anfield club on alert.