Seeking to add to his tally of 18 golds, Phelps topped the 200m butterfly semi-final times, winning his semi in 1min 55.17sec.
Pace Clark was second-fastest, winning the other semi-final in 1:56.27.
Phelps will have his first chance to secure his Rio berth in Wednesday's 200m fly final.
The 30-year-old superstar can become the first US man to make a fifth Olympic swimming team.
The disappointments for Franklin and Grevers underscored the difficulty of performing at such a high level for so long.
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Kathleen Baker finished second in 59.29 to put herself on the road to Rio.
Franklin faced an uphill battle after posting the seventh-fastest semi-final time, and the 21-year-old couldn't produce a miracle from lane one.
Her time of 1:00.24 left her 1.22sec off the pace.
"It's going to be really hard not to be in that," Franklin said, although her Olympic quest isn't over.
Less than half an hour earlier, she eased into the final of the 200m freestyle with the fourth-fastest time in the semi-finals, which were led by the indomitable Katie Ledecky.
Grevers, the men's 100m back gold medalist in 2012, came up just short, finishing third in a scintillating final won by Ryan Murphy in 52.26sec.
David Plummer grabbed the second Rio berth in 52.28sec, with Grevers out in the cold in 52.76.
- Fresh faces -
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Townley Haas, 19, won the men's 200m free final in 1:45.66, just one one-hundredth of a second ahead of Conor Dwyer.
Although the ailing Lochte failed to qualify to defend
his 400m individual title, he was pleased to make a fourth Olympic team.
Lilly King booked her first Olympic berth in style, winning the 100m breaststroke in 1:05.20, the world's fastest time this year.
Katie Meili stuck with her to finish second in 1:06.07, which put her inside the top five in the world this year.
While the pre-trials spotlight was on America's established stars, led by Phelps, three days of competition have produced a bevy of fresh faces on the Olympic roster.