The move is pending approval of university president Michael Crow, who is expected to return from a trip to China by the end of the week.
Edwards, 63, became a serious candidate for the vacancy on Tuesday, though he had not interviewed as of Wednesday.
According to ESPN, Edwards is slated to interview this weekend with athletic director Ray Anderson.
"I think I owe (Anderson) a conversation where we can communicate and sit down and talk," Edwards said on the Doug & Wolf Show in Phoenix. "It's for both of us because we have so much respect for each other. I think you have to respect people in life. You owe them a conversation, let's sit down and talk."
As an NFL head coach, Edwards owns a 54-74 regular-season record. He was 39-41 with the Jets, who reached the playoffs three times in his tenure.
Edwards' only college coaching experience was from 1987-1989 when he was the defensive backs coach at San Jose State.
Graham was 46-31 in six seasons for Arizona State. He went 18-19 in the final three seasons but guided the school to a bowl game for the fifth time in sixth seasons this year.
Graham has three seasons left on a contract that runs through the 2020 season and reportedly is owed at least $11 million.