Given, who was a schoolboy when Republic last played at the European Championships in 1988, said today: "It's a big two games home and away next month and we've got a real, real chance of qualification. We've got a very good squad with Ireland now and a real possibility to get to Poland and Ukraine next summer.
"That would be fantastic if we could get back there [to the finals of a European Championship]." Ireland beat England in the group stage at the 1988 finals but have fallen short of qualifying since then.
Given's selection for next month's play-off against Estonia was never in doubt, but Trapattoni raised some eyebrows by taking a chance on the fitness of striker Shane Long when naming his squad.
West Bromwich Albion's Long was initially ruled out for six weeks after sustaining a knee injury following a challenge from Aston Villa full-back Alan Hutton in last Saturday's Premier League game at Villa Park.
But Trapattoni has nonetheless included him for the two play-off games, in Tallinn on 11 November and the return in Dublin four days later. Villa goalkeeper Given has won a record 118 caps for Ireland and made his debut as far back as 1996, but he is not thinking about handing over the gloves just yet.
And Given is certainly not thinking about bowing out at EURO 2012. He told Absolute Radio: "I don't know, we'll see, but we'd better get there first you know. We don't want to get ahead of ourselves."
Given has dismissed his prospects of becoming the world's most capped player - he would need to play on for many years to pass the current mark of 178 - but stressed that the numbers game is of little interest to him.
"I'm not in it to make loads of caps," he said. "I'm in it to try to do well for my country and give something to them. If I'm good enough to play then I should play and if I'm not then it's time for me to step aside and someone else to come in and play. I'll make that decision when the time is right."
Trapattoni also selected Wigan Athletic midfielder James McCarthy and Stoke City striker Jon Walters, and the Italian manager told the Football Association of Ireland website: "There is no room for complacency as we prepare for our two games next month.
"We will remember that Estonia have as much right to be in the play-off as ourselves. They have had some very good results in their group beating Serbia and Slovenia. They also caused Italy a lot of problems. These games will be a huge challenge for us, and we must maintain the correct mentality from start to finish."
Republic of Ireland squad:
Goalkeepers: Shay Given (Aston Villa/ENG), Keiren Westwood (Sunderland/ENG), David Forde (Millwall/ENG)
Defenders: John O'Shea (Sunderland/ENG), Stephen Kelly (Fulham/ENG), Stephen Ward (Wolverhampton Wanderers/ENG), Richard Dunne (Aston Villa/ENG), Sean St. Ledger (Leicester City/ENG), Darren O'Dea (Leeds United/ENG), Paul McShane (Hull City/ENG)
Midfielders: Glenn Whelan (Stoke City/ENG), James McCarthy (Wigan Athletic/ENG), Seamus Coleman (Everton/ENG), Keith Andrews (Ipswich Town/ENG), Keith Fahey (Birmingham City/ENG), Darron Gibson (Manchester United/ENG), Andy Keogh (Leeds United/ENG), Damien Duff (Fulham/ENG), Aiden McGeady (Spartak Moscow/RUS), Stephen Hunt (Wolverhampton Wanderers/ENG), Liam Lawrence (Portsmouth/ENG)
Forwards: Robbie Keane (Los Angeles Galaxy/USA), Kevin Doyle (Wolverhampton Wanderers/ENG), Shane Long, Simon Cox (both West Bromwich Albion/ENG), Jon Walters (Stoke City/ENG)