The 31-year old joined Newcastle from Nottingham Forest last summer in a deal worth £20million with Elliot Anderson heading in the opposite direction for £35m.
However, Vlachodimos has only managed one appearance - as a substitute for Martin Dubravka in the Carabao Cup - and will slip further down the pecking order with the arrival of Aaron Ramsdale from Southampton on an initial loan.
Sevilla are exploring a loan deal of their own with Newcastle for Vlachodimos and know the Greece international is keen to move to gain more game time and get his career back on track.