Nine players qualified automatically for the team at the conclusion of the PGA Championship on Sunday, but a majority of the questions at a news conference at Valhalla Golf Club were about Woods, who missed the cut and was visibly hurting during the second round.
"I think it directly comes from Tiger," Watson said. "How he assesses himself. He has a directness to him. I'm direct and he's direct. The main thing is I can't assess his medical condition. And I really can't assess how he is playing. It's really going to have to come from him.
"I don't make this comment loosely. He is Tiger Woods and he brings a lot to the team if he has the ability to play and he is healthy. He brings a lot to the team. And I would be a fool not to consider him."
Watson reiterated his long-held stance that Woods must be healthy and playing well. The problem is that Woods is not scheduled to play between now and the Ryder Cup, which begins on Sept. 26. Watson is scheduled to make his three at-large selections on Sept. 2. He could pick Woods and replace him closer to the Ryder Cup if he is not fit.
Asked if he could trust Woods to give him a straight answer, Watson said, "Absolutely."
Later, Watson said, "I trust Tiger to give me the straight skinny. I trust him inherently."
Watson acknowledged that the United States will be an underdog for the matches at Gleneagles in Scotland. Bubba Watson, Rickie Fowler, Jim Furyk, Jimmy Walker, Phil Mickelson, Matt Kuchar, Jordan Spieth, Patrick Reed and Zach Johnson all qualified, but Kuchar is the last player in that group to win this year, and that was in April.
And Kuchar withdrew prior to the PGA Championship with a back injury, while Jason Dufner, who was bumped out of the last spot, withdrew during the first round with a neck problem that he said requires rest. The team has just three Ryder Cup rookies in Walker, Spieth and Reed, but none of the players outside of the top 10 -- Ryan Moore, Brendon Todd, Keegan Bradley and Webb Simpson, for example -- have made overwhelming cases.
Both Todd and Moore had a chance to make the team on their own Sunday, especially given Johnson's 70th-place finish, but were unable to put together a good enough final round.
"I think the European team is the stronger of the teams right now, on paper," Watson said. "But I have extreme confidence in the players that we have on our team and their motivation to go out and win the cup back. It doesn't leave me any question that we're capable of winning the cup."
The lone U.S. victory at the Ryder Cup since 1999 came at Valhalla in 2008 under captain Paul Azinger, who said Saturday on Golf Channel that he feels Woods should not play in the Ryder Cup.
"I just don't see how you can pick him at this point," said Azinger, an ESPN analyst. "I think Tiger actually will call Tom and say, 'I want to bail you out of this. I am just not quite ready.' That is what I feel Tiger will do and make Tom Watson's decision easier for him. With all due respect to one of the greatest players of all time, I would say you have to pass him up this year."
Woods has had his worst year as a pro in 2014, playing just eight worldwide tournaments, withdrawing from two and missing the cut in two others. He played just four events before having back surgery on March 31, then returned for four more, having to withdraw during the final round of the WGC-Bridgestone Invitational on Aug. 3 due to a different back issue.
He attempted to play at the PGA but did not look good, shooting two 74s and saying Friday that he suffered a setback on the driving range and attempted to play through it. Woods has expressed his desire to play in the Ryder Cup all along. His agent, Mark Steinberg, declined to comment Monday.
"He is Tiger Woods," Watson said. "He brings something to the team in a big way. He's been really good in the team room recently and he's a factor with the players. I know that for a fact. He's a very positive influence on the players.
"But the most important thing is, can he play? Can he physically play, and is he playing well, get back to those two points? Like I said, I'll monitor that situation in the next three weeks."
There has been speculation that Woods could add a European Tour event before the Ryder Cup -- he is not eligible for the FedEx Cup playoffs on the PGA Tour -- and the Wales Open (played at the 2010 Ryder Cup venue) is the week before the Ryder Cup.
Watson declined to be specific about what he expects in that regard.
"All I can do, as I said, is I've got to monitor his situation and during my conversations with him," said Watson, who added that in a conversation with Woods, the golfer told him, "I want you to pick me."
Woods has played on just one winning team in seven appearances and has an overall Ryder Cup record of 13-17-3. He is 4-1-2 in singles, with the only loss coming in his first appearance in 1997. Both ties were conceded matches after the competition had been decided.