The matches are September 28-30 outside Paris.
Mickelson has only been on three winning teams - at Brookline in 1999, Valhalla in 2008 and two years ago at Hazeltine.
Furyk will announce his final captain's pick next Monday to complete the 12-man team.
Mickelson had qualified for every team since 1995, a streak that ended this year when he finished No. 10 in the standings.
Mickelson has the second-worst Ryder Cup winning percentage of Team USA's top-10 all-time points scorers (47.7%, 18-20-7 overall record), edging out Tiger Woods, who has a disappointing 13-17-3 overall record (43.9%).
But the European captain insists the 38-year-old is "the heartbeat" of European teams and it would be like a football team going into a big match without their captain.
"It's incredible, it really is, to look back at the start of the year and now to have accomplished a goal like that", Woods, 42, said (via the AP).
The 25-year-old laced his drive, but it came to rest less than 5cm into the medium rough at the end of the fairway and although he was only 185m from the flag, his second shot was impaired sufficiently by the rough to leave it well short and in the hazard. And lo and behold, I'm a part of this team.
For the second straight week in the richest part of the PGA Tour season, DeChambeau took down one of the strongest fields of the year by playing his best golf on the weekend to win the Dell Technologies Championship, becoming only the second player to capture the opening two playoff events in the FedEx Cup. DeChambeau, Mickelson and Woods were the next three in the USA standings behind the eight players who earned automatic spots after the PGA Championship. "I'm very delighted with this team and I'm very proud to be captain for these 12 players".
Heading into a press conference to announce three of his four wildcard picks, Furyk already had a ridiculously talented and steely collection. I got off to a great start this year. The foursome has a combined two top-10 performances on the PGA Tour since the U.S. Open in June, and Garcia in particular has been lousy. For the continent to wrestle back the trophy these wildcards will be under pressure to repay the faith shown by their captain.
"Once I was able to make that putt on 14 it pushed me forward".
"He has been instrumental, not only in helping the captains before me, but me as well", Furyk said.
Ton Finau, Xander Schauffele and Kevin Kisner are all in contention to join the aforementioned trio and automatic qualifiers Brooks Koepka, Patrick Reed, Rickie Fowler, world no 1 Dustin Johnson, Jordan Spieth, Webb Simpson, Justin Thomas and Bubba Watson. He knows what I can bring to the team. "And naming these three players, that's what we've done".