Europe's Ryder Cup captain Jose Maria Olazabal says his side can overturn the United States' 10-6 lead, insisting: "It's not over. Simple as that."
As holders, Europe need to score eight points from Sunday's 12 singles matches to retain the trophy while the US need four-and-a-half points to reclaim it.
"We have a tough task ahead, but I believe it's not over," said Olazabal.
"That's what I learned from Seve (Ballesteros), and that's what I'm going to pass to the players."
The Americans have never lost more than a two-point advantage on the final day since the competition began in 1927 but Olazabal remains upbeat that his side can create history in Illinois.
The Spaniard said: "Things have not gone our way, especially on the greens, but I believe that momentum will come our way.
"We are trailing by four points and we have to put the players that are playing well up there. If we want to win this match we have to take the big guys down."
In the final day of the Ryder Cup at Medinah Country Club, Luke Donald will play Bubba Watson in the opening singles match as Europe set about their fightback.
Olazabal has put four of his most experienced players at the top of the order with Ian Poulter second out against Webb Simpson.
Rory McIlroy faces Keegan Bradley and Justin Rose takes on Phil Mickelson.
World number two Tiger Woods is the last man out for the United States, against Francesco Molinari. Woods faced the Italian two years ago at Celtic Manor, winning 4&3.
Paul Lawrie, who is fifth out against Brandt Snedeker, played in the same team as Olazabal when the US produced the biggest comeback in Ryder Cup history, overturning a 10-6 deficit to win at Brookline in 1999.
The Scot was one of only three Europeans who were victorious on that final day when the US picked up eight-and-a-half points.
And he said: "There's still 12 singles matches. We can all win and why not?
"It's certainly not over. It would be nice to make a big comeback, so we'll see what happens.
"It's always possible. Until it's impossible to do it, then you fight on, certainly for Jose Maria this week. No-one is going to be giving up."
Europe will play in the blue and white colours favoured by the late Ballesteros, who died in May 2011 after a long battle with a brain tumour.
Fellow Spaniard Olazabal was his partner for much of his Ryder Cup career and the players will have the image of Ballesteros on their bags as well as his silhouette on their shirts.
"We have Seve's blue and white, Seve on the bag and his right-hand man as our captain hopefully seeing us home," said Poulter, who has won all three of his matches so far at Medinah.
To win back the trophy they lost on Welsh soil two years ago, the US need to reach 14½ points but, although that score is tantalisingly within reach, captain Davis Love III is taking nothing for granted.
"We're excited about our position but we know, as Ollie said, it's not over yet," Love commented.
"All you can do is just go win your match and you can't think about the outcome of the day and what the other guys are doing.
"I don't want to talk about how many we need or what it's going to take to win, we just want to go play and we got one more day, for the next two years, to play Ryder Cup matches and they're going to be excited to come out and play."
The Amercian's famous Brookline win 13 years ago is also serving as a cautionary tale for Matt Kuchar, who takes on world number four Lee Westwood in the singles.
"There's a lot of memories from '99, and I think we have to know that that sort of deficit is overcomable," Kuchar said.
"We have to go out and still play some really good golf and make sure that we don't let a comeback like that happen to us."
Sunday's singles pairings (US first):
Bubba Watson v Luke Donald - 17:03 BST
Webb Simpson v Ian Poulter - 17:14 BST
Keegan Bradley v Rory McIlroy - 17:25 BST
Phil Mickelson v Justin Rose - 17:36 BST
Brandt Snedeker v Paul Lawrie - 17:47 BST
Dustin Johnson v Nicolas Colsaerts - 17:58 BST
Zach Johnson v Graeme McDowell - 18:09 BST
Jim Furyk v Sergio Garcia - 18:20 BST
Jason Dufner v Peter Hanson - 18:31 BST
Matt Kuchar v Lee Westwood - 18:42 BST
Steve Stricker v Martin Kaymer - 18:53 BST
Tiger Woods v Francesco Molinari - 19:04 BST