The 33-year-old Aston Villa man is yet to play for his club this season because of a persistent groin problem, but was put on stand-by for Wednesday night's friendly with Poland at the Aviva Stadium when Trapattoni named his latest squad last month.
However, after suffering a minor setback in his attempt to return to full fitness, Dunne has not been asked to travel to Dublin, with Trapattoni telling him to get himself fit for the forthcoming competitive clashes with Sweden and Austria which could go much of the way to deciding Ireland's fate.
Trapattoni said: "I spoke with him and I said 'okay. He wants to come back, he wants to be here', but I said to him that at this moment, I don't want to make trouble with the club.
"We need him for the matches and it will be important that he gets through these medical tests.
"I hope he can be ready for the matches."
Villa are currently engaged in a tooth-and-nail battle for Premier League survival and manager Paul Lambert's need is perhaps even more pressing than Trapattoni's.
Toronto central defender Darren O'Dea will also be missing after being excused duty at the request of his club, who are currently in the middle of the preparations for the new MLS season.
Trapattoni is likely to use the game to blood some of his less experienced players, with Derby duo Richard Keogh and Conor Sammon expected to have some involvement.
The Italian is delighted at the way his pool has increased in size since he took over as manager almost five years ago, and is keen to have a look at some of the newcomers before turning his attention back to the World Cup qualifying campaign.
He said: "They have good quality to come into our team and I am very, very anxious to see them. It will be important.
"They can give us their enthusiasm and commitment and their technical quality, I am sure they can play well."
Trapattoni's initial squad of 27, which included Dunne, has been whittled down with Reading's Alex Pearce taking O'Dea's place.
However, he lost two more men on Saturday when Norwich's Anthony Pilkington withdrew after damaging a hamstring in his side's 0-0 Premier League draw at QPR, and Spartak Moscow winger Aiden McGeady cried off with an ankle problem.
While the result on Wednesday evening may not be the most important aspect of the game even for a manager who insists there is no such thing as a friendly, Trapattoni will be keen to avoid a home defeat for the third successive match.
The Republic were thrashed 6-1 on their own park by Germany in October, sparking concerted calls for the 73-year-old's head which he only just survived.
They bounced back four days later to beat the Faroe Islands comprehensively in Torshavn, but they went down to a single goal against Greece at the Aviva in a November friendly.