Furyk will add a final pick Monday following the BMW Championship at Aronimink Golf Club.
It is inconceivable that 14-times major champion Woods could be left out after the form he has displayed in his comeback this year from a potentially career-ending back injury. Seven missed cuts from his past twelve starts, although a tie for 8th at the Open de France, which was played at the Ryder Cup's Le Golf National venue, might play in his favour.
Woods, a 14-time major champion, and DeChambeau, coming off this third career PGA title at last week's Northern Trust event, are each expected to receive "wildcard" picks from United States captain Jim Furyk on Tuesday.
DeChambeau, Mickelson and Woods were the next three in the USA standings behind the eight players who earned automatic spots after the PGA Championship.
DeChambeau, 24, had virtually guaranteed himself one of the captain's picks after storming to victory in the opening two legs of the PGA Tour's FedEx Cup playoff series, winning the Northern Trust and Dell Technologies Championship. Furyk will make three selections on Tuesday, and a fourth after the BMW Championship.
The 2018 Ryder Cup gets underway on September 28 and the buzz is building with both team captain's to reveal their wildcard picks this week.
"But also deep down I wanted to make the team".
"It's obvious that the one thing that has been missing is for our team to go over to Europe and win", said Mickelson, who has played on three winning US teams - but never overseas.
Rose bogeyed his first two holes and played his front nine in 1 over, but birdied four of his last six holes.
When Mickelson, 48, won in Mexico in the March to end a five-year winless drought, he thought it would be a certainty he'd be on the team.
Furyk said Woods will no longer serve as a vice captain. Finau is in strong consideration to become one of the four captain's picks for the U.S. Ryder Cup team. Woods was an assistant at Hazeltine and at the Presidents Cup last fall at Liberty National in a USA route.