The 31-year-old, who returned to training in December after undergoing successful treatment for testicular cancer, is in talks with Velez Sarsfield.
Gutierrez, who joined the Magpies from Real Mallorca in July 2008, is out of contract at the end of the current season and has not played a senior game for the club since October 2013.
Gutierrez, who has made 195 senior appearances and scored 11 goals for the Magpies, will be remembered on Tyneside for his seldom-seen Spider-Man goal celebration, and the fact that he was one of those players who stayed on after relegation in 2009 to drag the club back into the Premier League at the first attempt.