The Icelandic international has impressed this season for a hugely competitive Clarets side, and his performances have alerted other clubs in the division - with the Magpies, Leicester City and Southampton all reportedly keen.
The 2016 signing from Charlton has started every game since October under Sean Dyche, and has been a key figure for the club en route to seventh place.
TEAMtalk claim the club will try to tie Guðmundsson down to a new deal this summer, since he only has one year remaining on his current contract.
The Clarets can trigger an extension clause, and will surely take up the option to do that to try and fend off potential suitors. For the moment though, a fee of around £20m could be enough to prise him away.
Rafa Benitez is reportedly a big fan of the 27-year-old and would like to sign him up for next season to provide competition for the likes of Matt Ritchie, Christian Atsu and Kenedy if they can successfully loan him in again from Chelsea.
Leicester are likely to lose wantaway Riyad Mahrez in the summer and could turn to Guðmundsson to replace the Algerian. The Saints could also enter the race if they avoid relegation.
The ex-AZ Alkmaar player has scored two goals and contributed six assists across 29 league appearances for the Turf Moor outfit so far this season, and will be eyeing a strong finish to help the club hold seventh place.