The Football Association is looking into the shocking incident that occurred in the second half of a 2-2 draw between the two sides at Anfield on Sunday - before Suarez scored an equaliser in the seventh minute of added time.
Suarez, who was banned for seven games for biting an opponent during his time at Ajax, sunk his teeth into Ivanovic's arm after the players came together. He later made a public apology as the club said his conduct was "not befitting of any player wearing a Liverpool shirt".
Merseyside Police have spoken to Ivanovic and the Serbia international has said he does not want to press charges against the Liverpool striker.
Suarez has been fined an undisclosed sum by Liverpool though, and has asked the club to donate the money to the Hillsborough Family Support Group, which since May 1989 has been fighting for justice for the victims of the tragedy which resulted in the deaths of 96 Liverpool fans.
Suarez wrote on Twitter: "For my unacceptable behaviour yesterday the club has fined me today, I have asked the club to donate the money to the Hillsborough Family Support Group for the inconvenience I have created to the Liverpool fans and to Ivanovic."
Suarez's future at Anfield has been the subject of speculation since he bit Ivanovic but managing director Ian Ayre says the player signed from Ajax for £22.8 million in January 2011 will not be forced out of the club.
"Not at all," Ayre told the club's official website. "It affects his future in the sense that we have to work with him on his discipline - but Luis is a very important player to the club.
"He's a very popular player with his team-mates. As we keep saying, he signed a new four-year contract last summer and we'd all love to see him here throughout that contract.
"He's a fantastic player, top scorer and everything we'd want in a striker, so there's no change there. This is more about getting him back on the right track and it's largely down to Brendan now to work with him on that side of his character."
PFA chief executive Gordon Taylor says the organisation will offer its counselling services to club and player in an attempt to curb Suarez's apparent anger issues.
"There is no doubting his football ability, that's why it is so disappointing and embarrassing when he lets himself down," Taylor said.
"We have to work hard on anger management now. We have trained counsellors in this field and we will be offering their services to Liverpool and the player to try to improve matters."
In a statement to the BBC, Suarez's sponsors Adidas said the company "does not condone Luis Suarez’s behaviour. We will be reminding him of the standards we expect from our players."