Sessa, who last month kicked a ball angrily at a ball boy during an away game in Cordoba, was sent off after 25 minutes of the 3-0 defeat against Boca Juniors on Wednesday for a wild kick at Rodrigo Palacio.
Sessa went up to collect a cross, but as he did so, raised his foot and landed his studs in the Boca forward's face. He was sent off and gave away a penalty, missed by Martin Palermo.
Palacio played on after being treated for a cut on his forehead.
"The work we did before the game all went down the drain because we had to play with one man less," Velez vice-president Roberto Jimenez told the Telenoticias cable channel. "It was an unfortunate and unnecessary act."
"We have to analyse what has happened and take a decision with cool heads...It's an unacceptable situation for a club such as Velez."
Velez coach Ricardo La Volpe, himself a former goalkeeper, told reporters: "Only he knows why he did it. Sometimes humans make mistakes, that's what makes them human."
"We know we can't defend an act like this...He himself has to sit down and reflect and make the best decision."
The 34-year-old goalkeeper, who has been at Velez since 2001 apart from a brief spell in Spain with Las Palmas, is no stranger to trouble.
In 2002, he was banned for 10 games after grabbing a referee by the neck after being sent off. He also kicked advertising boards in fury as he left the pitch.
The following year he dropped his shorts in front of fans of rivals Racing Club after a penalty was awarded against his team.
Last October, he attacked team-mate Lucas Castroman, apparently angry at the way in which Castroman had just got himself sent off, in the tunnel.
This year, he slapped team-mate Maximiliano Pellegrino in the face during a Libertadores Cup match away to Internacional in Brazil but escaped a red card because the referee did not see the incident.
That was followed by what the Argentine media described as his day of fury in Cordoba when, in addition to the ball boy incident, he grabbed his genitals in front of a photographer.
(Writing by Brian Homewood in Rio de Janeiro)