The 26-year-old is likely to miss Saturday's Premier League clash with Wigan after suffering the knock following a challenge with Joleon Lescott against Manchester City.
Turner reveals progress has been good, but admits he is unlikely to make a comeback before the Gunners clash on 18th September at the Stadium of Light.
"Last week the ankle was very swollen, but towards the end of the week it settled down a bit and I got some rest over the weekend," he told the club's official website.
"I'm still getting pain from it, though, so hopefully it will settle down soon. There's not much more I can do at the moment except be patient and wait for it to sort itself out.
"The Arsenal game is probably the realistic target for me. It's hard to tell. It could settle down earlier or it could swell up again, it's hard to say."
Despite the unfortunate injury, Turner has no bad feeling towards City's Lescott following the challenge.
Both defenders slid in for the ball and Turner came off worse, but the Black Cats stopper says no-one was to blame.
He added: "I thought I was ahead of him and he thought he could win the ball - it's just one of those things."