The 33-year-old Spaniard, who also played for Liverpool, said in an Instagram post that it was "onwards and upwards" after receiving his first MRI scan results.
Enrique was diagnosed with chordoma, a rare type of tumour which usually grows in the base of the skull and spine.
But his formed behind his eye and he was told he could lose his sight.
Enrique retired due to a chronic knee injury in September 2017 while playing for Spanish second tier side Real Zaragoza.
He made 127 appearances for Newcastle after joining from Villarreal in 2007 for £6.3million and was signed by Liverpool for a reported £6million in the summer of 2011.
Enrique made 99 appearances for Liverpool before being released in May 2016.