The 22-year-old arrived at Goodison Park from Iraklis Thessaloniki in 2011 and has since made only 24 appearances for the club, scoring three goals.
The Greek frontman failed to make a single league appearance for Everton during the 2013-14 campaign, but made two starts during a loan spell at Championship side Blackpool.
And Spaniard Martinez believes Vellios will benefit from a move away from Goodison Park.
"I think it's been a disappointing season for himself but he's always had a terrific attitude and always a real desire to try to work hard for the team," Martinez told evertontv.
"He's probably in a position now where, at a very young age still, he needs a new challenge and a new environment to show what he can do.
"We wish him well and I'm sure every Evertonian will follow me with that."
Goalkeeper Mason Springthorpe, 19, has also left the club after his contract expired.
‘Even though it was late in the game, the Arsenal fans started singing my name. It felt like the moment I announced myself as a professional footballer’: Ex-Gunners star recalls scoring first senior goal as a teenager at Highbury
'I started to drive myself crazy': Former Manchester United midfielder explains why it all went wrong for him under Erik ten Hag
