Tardelli told an audience in Montecatini, with tongue slightly in cheek, that he thought the Republic of Ireland would draw with Croatia in their opening Group C fixture next Sunday night and then beat both reigning champions Spain and his native Italy to qualify for the quarter-finals.
Trapattoni was not so forthright in his own assessment of the group in Budapest, but admitted he would be happy if the scenario Tardelli envisages comes to fruition.
Speaking at the Ferenc Puskas Stadium ahead of Monday night's final friendly against Hungary, the 73-year-old said: "Marco is full of enthusiasm and dreams.
"But sometimes dreams come true. It is important we start with the first 90 minutes. After that, we can dream of continuing this way."
Skipper Robbie Keane too is keeping his fingers crossed that Tardelli has had some kind of premonition.
He said: "I hope it's a dream that comes true, to be honest with you."
Trapattoni will field his strongest side in the game and has already said he is 90% certain that the men who start in Budapest will also do so in Poznan next Sunday, when the Croatians stand between the Republic and a winning opening to their campaign.
To that end, he was delighted to be able to name keeper Shay Given and defender John O'Shea in his starting line-up with both men having struggled for fitness over the last 10 days or so because of knee and ankle injuries respectively.
He said: "Tomorrow determines next week. Given is okay, but I was waiting for O'Shea. He has no injury, but he hasn't played with us.
"Until this game, I don't know if he is 100%. That is the test for him, and I'm sure now he is ready. After the game, hopefully we can say he is okay and is 100%."
However, Trapattoni has indicated he is likely to withdraw some of his key players before the final whistle, and O'Shea could be one of them.
He said: "Maybe not 90 minutes. Maybe it is good he gets just 70 minutes. It is no problem. I am thinking about Croatia."
O'Shea will line up ahead of Given in a defence which will also include Richard Dunne, Sean St Ledger and Stephen Ward, with Damien Duff, Keith Andrews, Glenn Whelan and Aiden McGeady in midfield and Keane and Kevin Doyle in attack.
The Ireland manager has made a habit of announcing his team the day before a game, and makes no apologies for it, insisting that opposing managers already know enough about his players to make subterfuge pointless.
He said: "I watch DVDs and we have scouts at every game. Other managers watch and they see the games our team has played.
"It is not only the last 90 minutes. They know all about us, we know all about them."
The Republic go into the game on the back of a 13-match undefeated run, and it is one they are determined to extend to give themselves the perfect platform for the finals.
Trapattoni said: "This is the last game in preparation for the Euros. I would like to see the same performance, the same enthusiasm, the same fluidity in the game that we saw against Bosnia and also in training last week.
"Hungary are quite a young team with a very consolidated football tradition.
"I have already said the team that starts tomorrow will be the same that will start against Croatia, unless there are some unexpected circumstances."
Tonight's press conference in the former Nep Stadium brought back fond memories for the veteran Italian, who played there as an 18-year-old back in the 1950s.
He said: "I remember the great Honved. (Gyula) Grocsis, (Nandor) Hidegkuti, Ferenc Puskas and (Zoltan) Czibor - they were fantastic. This was before the Cold War and the Iron Curtain.
"They were like Barcelona are now. Honved were unbelievable."
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