Benitez has made defensive reinforcements one of his top priorities this summer after seeing Steven Taylor and Fernando Coloccini leave St James' Park since the end of last season.
Newcastle have seen several bids for Blackburn centre-half Grant Hanley rejected and they are now ready to move on to other targets after being frustrated in their pursuit of the Scotland international.
Amorebieta's versatility - he can play at centre-back and left-back - makes him an attractive proposition for Benitez as the Newcastle boss is also in the market for a left-back.
The 31-year-old has spent time on loan at Middlesbrough during the past two seasons, making 22 appearances for Aitor Karanka's side who were promoted to the Premier League last season and his experience of playing in the Championship has caught the eye of Benitez who is looking to build a side capable of winning promotion back to the Premier League next season.
Amorebieta joined Fulham from Athletic Bilbao in 2013 and has made over 50 appearances for the West London side.
The Venezuelan international spent 10 seasons at Bilbao prior to his move to Craven Cottage and helped the La Liga club reach the finals of the Europa League in 2012 and Copa del Rey in 2008 and 2012.
Amorebieta is entering the final year of his contract at Fulham and they could be willing to let him leave this summer rather than risk letting him run down his contract.