Arokodare and Mundle were two of four Premier League players who received racist abuse on social media last weekend, with Burnley’s Hannibal Mejbri and Chelsea’s Wesley Fofana being similarly targeted.
The UK Football Policing Unit (UKFPU) is investigating the incidents and said it will “do everything possible to identify those responsible and bring them to justice”.
Police Scotland has also said it was investigating after Rangers players Emmanuel Fernandez and Djeidi Gassama shared screenshots of racist abuse aimed at them on their social media accounts after Sunday’s 2-2 draw at Livingston.
Anti-discrimination group Kick It Out said reports of online abuse are at record levels, increasing by a third compared with the same point last season.
Mike Ankers, UKFPU deputy director, told BBC Radio 5 Live the police unit has had an increase of about 115% in the number of reports.
“I don’t know really why, but I repeat – we don’t want to accept that,” added Le Bris.
“We have to have a clear position and I think it’s the case here at Sunderland, we want to defend values and when someone, a group of people, want to threaten those values it’s important to react.”
Edwards added: “There have been a number of these instances, and very high-profile over this last week, which is really sad.
“I have been a head coach and a manager a number of times when my players have had this and the sad thing about it is all of them have had it before.
“They say, ‘I’m OK, I have had it before’. I think that is what is difficult. That is tough to hear.”
Edwards said Arokodare had trained this week and, despite being affected by the racist abuse, was “all right to play” in his side’s home game against Aston Villa on Friday (20:00 GMT).
However, Le Bris said Mundle “might be sidelined for a few weeks or months” with an injury and would miss Sunderland’s game at Bournemouth on Saturday (12:30).
Responding to the Fofana and Mejbri abuse at the weekend, a spokesperson for Meta, which owns platforms including Instagram, told BBC Sport: “No-one should be subjected to racist abuse, and we remove this content when we find it.
“No one thing will stop racist behaviour overnight, but we’ll continue working to protect our community from abuse and co-operate with police investigations.”