Two Super 14 teams - the New South Wales Waratahs and the Queensland Reds - are still in the battle to sign Tuqiri, as well as one Australian Rugby League club, the South Sydney Rabbitohs.
However, ARU CEO Gary Flowers, who made an original offer to Tuqiri in November but withdrew it when it was not accepted, hopes to end the protracted signing saga as soon as possible.
"We always said it would go back to the normal contracting process and that involves two state unions - Queensland and NSW," he told the Daily Telegraph.
"We will be bringing it to a conclusion now as quickly as we can."
The Waratahs tabled their offer for Tuqiri last week while the Queensland Reds put in an offer for the 27-year-old on Tuesday.
Tuqiri's current Super 14 coach at Waratahs, Ewen McKenzie, said the intense media speculation had taken its toll on the Wallabies winger, and had distracted his team-mates, but he hoped the player would take up their offer.
"I would be lying if I said it hasn't been a distraction," McKenzie said.
"Every spare moment he is probably talking with his manager and talking with interested parties.
"As far as I am concerned, we have done our bit. We did it early last week so we are waiting."
The star winger has admitted he has found the media speculation surrounding his future very stressful and has even had to refute claims that he doesn't want to stay in rugby after entertaining offers from rugby league teams.
"I don't think there is evidence to say I don't want to play rugby," Tuqiri said. "I'm playing the game now, I want to play every week and I want to win.
"It's actually been quite stressful. I just found this time there was more media attention with the spotlight on you everywhere you go."
He added: "I probably didn't realise how much this would impact on me. But this game has given me the profile that I have now, and you have to roll with it."