The England Under-21 international was signed when Kieran Richardson was sold to Fulham in August and the Tottenham player has impressed in the North East.
Now O'Neill admits he would like to finalise an extended stay for the former Leeds United trainee.
"There is that possibility," said the Sunderland boss when asked if Rose could stay beyond the end of his current season-long terms.
"He wants to remain at the football club and that is encouraging.
"If he went back to Tottenham, particularly with [Benoit] Assou-Ekotto getting fit again, the chances are he wouldn't play that often, so it's certainly worth a look at anyway.
"He would love to stay here. He's got on terrifically well with everyone. The players have taken to him, everybody has taken to him. Most importantly, the crowd have taken to him and rightly so because he has been absolutely outstanding.
"I always felt he had that potential, but he's playing remarkably well. He's been really excellent for us and he's improving as well.
"I don't really want to shout it out because we want to try to keep him at the football club, but Danny has been splendid ever since he arrived," said O'Neill.
"As his fitness improves moving forward, you feel as though he is very capable of playing in that (midfield) position.
"We wanted to try him there beforehand, but he's been so decent at left-back that I didn't want to change him until he gets the requisite amount of fitness that you probably need to play in midfield. But he was excellent when he moved there."
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