However, the Queensland Reds star will have an MRI scan on his left knee that is likely to end his hopes of winning back his Wallabies' jersey after three injury-plagued years.
A lump on the joint line of Tune's left knee is the problem and the scan will determine if it needs to be removed.
"Even if I decide to tough it out and not have the surgery, I'd still struggle," Tune told The Australian newspaper.
"Either way, I think I'm extremely unlikely to be right for the tour."
It is yet another setback for the 28-year-old, who returned to the Reds' starting line-up for the second half of the Super 12 campaign this year.
South Africa-born winger Clyde Rathbone is also doubtful with tendinitis in his left knee.
Although Rathbone played three Tri-Nations Tests, he revealed the last time he had been fully fit going into a game was April 2, when he damaged his medial ligament in the Super 12 match against the Blues in Auckland.
So badly have his form and confidence been dented by injury that Rathbone is determined not to play again until he is 100 per cent right, even if that means missing the European tour.
It's a decision that has the full support of Wallabies' coach Eddie Jones.
"To his great credit he played those last two Tests (against the Springboks in Perth and the All Blacks in Auckland) only 75-80 per cent fit but we're very keen to get him back to 100 per cent which is where he needs to be because he relies so heavily on his power and speed," Jones said.
With Stirling Mortlock being rested, the tour is Rathbone's best opportunity to stake his claim for the outside centre position where he made his Test debut against Scotland last year.
Since then, however, he has played most of his Test rugby on the wing.
"I've still got some time to experiment and if I can get a crack at 13 I can at least make an informed judgement on which spot suits me best," Rathbone said.
"It's hard to say at this stage but long term my best position might be at centre."
Number eight David Lyons, who is recovering from a groin reconstruction and a hernia, is also struggling to meet Tuesday's selection deadline.
"Unless he's absolutely 100 per cent, we won't take him," Jones said.
"He's progressing really well but he's another of those guys who has played consistently since 2000 and we want him back to 100 per cent.
"We're taking a pretty hard approach in that regard."
The touring party, which will be made up of 31 Wallabies and seven or eight players initially designated just for the Australia A game, is likely to include four scrum-halves in George Gregan, Chris Whitaker, Matt Henjak and Brett Sheehan, who will make his debut for the NSW Waratahs against the Czech Republic in Prague.