Only Toulouse have managed a hat-trick of Heineken Cup titles but back-to-back wins in 2001 and 2002, allied to their historic semi-final penalty shoot-out against Cardiff Blues earlier this month, have left the Tigers on the brink of matching Toulouse's achievements.
Murphy is in fine form, as evidenced by his man-of-the-match display in last weekend's Guinness Premiership final win over London Irish.
"It is the premier European club competition and we would dearly love to win it for a third time," admitted the Kildare man, who is taking on the captaincy duties for the sidelined Martin Corry.
"The Heineken Cup means a tremendous amount to this club - we have been there or thereabouts in most tournaments - and we have a great relationship with the tournament.
"I have played in three Heineken Cup finals, winning two and losing one, so I know all about both the elation of winning and the depression of losing.
"It really is a bitter pill to swallow when you don't win and we will be doing everything we can to get that win.
"I have enjoyed some great moments with some great Leicester teams but all that is in the past, it is all done and dusted.
"When the time comes I'm sure I will look back on those occasions with great affection but at the moment they don't mean a thing - it is all about Saturday and Leinster."
This is Murphy's testimonial year with Tigers. He has spent 13 seasons at the Welford Road club, winning a bucket-load of trophies and medals.
But the desire for more silverware burns brightly inside everyone of Richard Cockerill's players, including Murphy's Kildare compatriot Johne Murphy.
"We want to reward our fans who spend so much time and money supporting us and the best way of doing that is to win silverware and at Leicester we are completely aware that you have to earn everything, you cannot take anything for granted," Geordan added.
"But make no mistake, we will be going into the game as the underdogs. Before the semi-finals everyone was saying whoever won the all-Irish semi-final would go on to win the final so that has to be Leinster as the favourites."
Before taking part in a flurry of media interviews earlier this week, Murphy was on the phone to his good friend, Leinster's talismanic centre Brian O'Driscoll, to exchange some pre-match pleasantries.
The pair enjoyed a light-heared debate over which side should start as favourites, with O'Driscoll arguing that his men will be underdogs.
Murphy added: "We had a good chat and a laugh. It was good-natured stuff about what is going to happen next week.
"I told him to give over and suggested everyone back home already felt they had the trophy sewn up.
"Paddy Power have paid out already back in Dublin! It was surreal I suppose.
"Having a chat with a great mate and then knowing I'll be facing him in the biggest club cup final of the season.
"But I know him and all the Leinster lads so well. I won't need to do any special computer analysis."
Murphy has scored three tries in Leicester's eight-match march to the Edinburgh final, including a score in that dramatic semi-final defeat of Cardiff.
He has a 50-50 European record against Leinster, having helped Leicester to three wins in six matches against the Irish province since 1999.
Crucially, Tigers won the sides' 2005 quarter-final meeting at Lansdowne Road and their last clash (25-9 at Welford Road) in January 2008, but their long-serving number 15 rates Michael Cheika's current crop very highly.
"Leinster have always had a world class back-line, now they also have the forward power to maximize those strengths and flanker Rocky Elsom has been possibly their stand-out player this season, a truly tremendous signing.
"They have got strength across the board and you cannot single out any one individual.
"Brian O'Driscoll is one of the best centres in the world but if you worry and concentrate on just one player then you will only create space and opportunities for the likes of Gordon D'Arcy, Luke Fitzgerald and Rob Kearney.
"On top of all that, Leinster have never won the Heineken Cup and that will make them an extremely hungry side and give them the desire to beat us.
"There is a feel good factor in Irish rugby after the Six Nations and that is fantastic for the country but on Saturday it will only be all about Leicester and Leinster and, if we come out on top, I hope they will not be too disappointed.
"We have made a lot of finals and not won them all and that is very disappointing so it is something we want to address and we will go and give it our best shot.
"And me as captain is not that big a deal - it's just a 'c' after my name.
"It's not rocket science. I'm not doing anything exceptional with the lads. I'm not saying anything special. I'm generally losing the coin toss. That's all."