After joining Barca from Atletico Madrid for €120million in July 2019, Griezmann scored 15 goals in 48 appearances in all competitions last season – his worse return since 2012-2013 with Real Sociedad.
The 29-year-old attacker averaged one LaLiga goal every 285 minutes for the Blaugrana, way below his best return of one every 112 minutes for Atletico in 2014-2015.
He has been vocal about his frustration at playing on the right wing since Ronald Koeman took over as Barca coach in August and only played eight minutes off the bench in the 3-1 Clasico defeat to Real Madrid on Sunday (AEDT), having been an unused substitute in the midweek thrashing of Ferencvaros.
World Cup-winning left-back Lizarazu believes Griezmann, who is reportedly set for a new contract at Camp Nou, should look to move on before his situation deteriorates any further.
"Griezmann is great. He's a very, very good player and he's fantastic for France. But he isn't at all for Barcelona. This is a real problem," Lizarazu said.
"He went to Barcelona with a dream of playing with Lionel Messi, but unfortunately there is no technical relationship between the two.
"He cannot find his position, he can no longer player football with Barca. It's a crisis of confidence.
"At some point it is better to cut off your hand than your arm. Do not hesitate to leave when you cannot find your place.
"It's been 18 months, not just two... the situation is really critical."
Griezmann and Messi are yet to create a single chance for one another in LaLiga this season, though the Barca captain has provided one key pass each to Francisco Trincao and Gerard Pique, two to Ansu Fati and three to Philippe Coutinho.
They have also combined on just 23 passes, with Messi having shared more with Fati (35), Frenkie de Jong (39), Jordi Alba (47), Sergi Roberto (54), Coutinho (55) and Sergio Busquets (109).