Needing only a further 3-1/2 points to keep the trophy after establishing an overall lead of 14-8 earlier in the day, the Americans led in five of the 12 matches, trailed in three and were all square in the remainder.
No team has ever triumphed in the Presidents Cup after coming from behind going into the last-day singles, leaving the holders in a commanding position to win the Cup for an eighth time in 10 editions.
The biggest obstacle facing the Americans could be the weather - heavy storms are forecast for later in the day with provision already made for any eventual Monday finish.
In the first match out, Hunter Mahan was three up on Japan's Hideki Matsuyama after 10 holes, having already recorded four birdies.
PGA Championship winner Jason Dufner was one up on Zimbabwean Brendon de Jonge after six holes while Bill Haas led by the same margin against Australian Adam Scott, after five.
Zach Johnson was two up on South African Branden Grace after four holes and Matt Kuchar was one up on Australian Marc Leishman, after four.
Posting blue numbers on the board for the Internationals were Australian Jason Day, two up on Brandt Snedeker after nine holes, and South African Ernie Els, two up on Steve Stricker after seven.
Amid sporadic rain showers, some of them heavy, the Internationals were also ahead in the 10th match out where South African Charl Schwartzel was one up Keegan Bradley after two holes.
World number one Tiger Woods was all square with South African Richard Sterne after four holes.
Earlier, the U.S. had opened up a commanding 14-8 lead after winning two of the remaining four matches in the delayed foursomes session while halving one.
Haas and Stricker wrapped up a comfortable 4&3 victory over Masters champion Scott and Japan's Matsuyama while Simpson and Snedeker came from three down after 12 holes to beat Louis Oosthuizen and Schwartzel one up.
Phil Mickelson and Bradley halved their match with Day and Canada's Graham DeLaet before South Els and de Jonge inflicted the first defeat of the week for Woods and Kuchar, winning one up
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