US Captain Jim Furyk will name his final wildcard pick next Monday after the BMW Championship.
For Woods, it'll be his eighth appearance and first since 2012.
He has yet to win since making his comeback but was in contention at both the Open Championship and the PGA Championship while Phil Mickelson also won his first PGA Tour event since 2013 at the WGC-Mexico Championship in May.
"It's going to be a great challenge because we know how strong the European side is and how well they play at home, but it's a wonderful chance, an opportunity for us to do something I haven't done or been a part of in my career, and would very much like to", Mickelson said.
"I was leaking oil a little bit", said DeChambeau, who after bogeying the 13th steadied the ship by sinking a five-footer to save par at the next.
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Mickelson hasn't missed a Ryder Cup squad since the early 1990s having played in a record 11 consecutive tournaments.
That hasn't always been the case for Woods, who is 9-16-1 in Ryder Cup team matches.
It is only the identity of his final wild card, to be named after the third FedEx Cup play-off event on September 10, which will be causing Furyk a headache with the likes of Xander Schauffele, Matt Kuchar, Tony Finau, Kevin Kisner and Zach Johnson in the frame.
'I got the call from Jim, asking if I would be a vice-captain and I said: "Absolutely, I'll do anything to help you" but deep down I wanted to make the team. "I get chills just thinking about it". "It's a well-rounded team, and we'll go to work, get set to go".
Mickelson had qualified for every team since 1995, a streak that ended this year when he finished No. 10 in the standings. Among the players he's considering for his four at-large selections, include Ian Poulter, Sergio Garcia, Henrik Stenson, Paul Casey, Rafa Cabrera-Bello, Thomas Pieters and last week's victor in Demark, Matt Wallace, a multiple victor on the European Tour this year.
Phil Mickelson and Bryson DeChambeau also received at-large picks and will join qualifiers Brooks Koepka, Dustin Johnson, Patrick Reed, Justin Thomas, Bubba Watson, Jordan Spieth, Rickie Fowler and Webb Simpson.