Mickelson had qualified automatically for the U.S. team stretching back to his debut in 1995, making 11 consecutive appearances in the biennial event.
The top eight on a points list after the PGA Championship on Sunday will make the team automatically, while captain Jim Furyk will select four others.
Five-times major champion Mickelson arrived at the PGA Championship 10th in the standings, and did not even come close to making the halfway cut, shooting four-over-par 144 to miss out by four strokes at Bellerive.
Though he has not been in good form recently, the 48-year-old will probably still be picked by Furyk, who has a few more weeks to ponder his choices.
Less likely to make the team is ninth-ranked Bryson DeChambeau, who missed a four-foot putt at his final hole on Saturday to miss the cut by one stroke.
Twelfth-ranked Matt Kuchar also missed the cut, but 11th-ranked Xander Schauffele and 15th-ranked Kevin Kisner advanced.
Eighth-ranked Webb Simpson, the vulnerable man in the qualifying hot seat, also advanced, so has a chance to take care of his own fate over the final two rounds.
But if he does not have a high finish, Simpson could be overtaken by Schauffele or Kisner.
Schauffele, equal 19th after two rounds, can pass Simpson with a finish of fourth or better.
Kisner looms a big danger too. In second place after 36 holes, he will automatically make the team with a first or second placing on Sunday, and perhaps even with a third.
Six of the top eight Americans in the standings made the cut on Saturday, with only fourth-placed Patrick Reed and fifth-placed Bubba Watson bowing out.
However, they are already locked in to make the team.
Seventh-placed Rickie Fowler is not yet a mathematical certainty but as he is only two shots from the lead before starting the third round, he is expected to pick up enough points to hold on to his spot.
(Reporting by Andrew Both, editing by Pritha Sarkar)
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