The incident took place during the second half of Sunday's 2-2 draw between the two sides at Anfield in the Premier League.
However, Suarez was not punished by referee Kevin Friend and went on to score a late equaliser to earn Liverpool a point.
"His behaviour is not befitting of any player wearing a Liverpool shirt and Luis is aware that he has let himself and everyone associated with the club down," Ayre told the club's official website. "We will deal with the matter internally and await any action from the FA."
The Football Association has confirmed that it will investigate the incident on Monday.