Greer's teammates talked, however, about being prepared to move on without the longtime veteran leader.
"We're just kinda keeping our fingers crossed with him," Payton said, explaining that Greer was still meeting with doctors Monday. "I spoke to him last night in the hospital.
"It's always difficult when someone like Jabari, who's been with us and been a staple of what we've done defensively, goes down with an injury like he sustained. It's a knee injury, and then I'll fill you in with the specifics [Tuesday]."
It was hard to deny the severity of Greer's injury when it happened during Sunday's 23-20 victory over the San Francisco 49ers.
Replays captured his left knee bending in an extremely awkward position as he landed after breaking up a pass in the first quarter. Trainers stabilized his leg with an air cast before Greer was carted off the field.
Teammates are obviously preparing for the worst-case scenario.
"That was a tough one ... just seeing how he is today [in the locker room] and knowing things are gonna be a little different around here," said receiver Robert Meachem, who said Greer visited the locker room Monday and seemed to be in good spirits, considering the circumstances.
Greer, who has been a full-time starter since he joined the Saints in 2009, was playing at a high level this season for a resurgent defense. He had one interception and 12 pass defenses, including two before he left Sunday's game.
Greer was replaced in the lineup by second-year player Corey White, a physical cornerback who has shown promise in his young career. White immediately stepped up with an interception in the second quarter Sunday -- though he wound up fumbling the ball through the back of the end zone for a touchback.
"Jabari's been great this year. He's played up to what he did from previous years and was playing at a very high level," Saints defensive end Cameron Jordan said. "For him to go down like that, you have to rely on the coaching staff to pick players on our team and to know that it's the next man up.
"We can't replace Jabari and the ... spiritual leadership he brings to the team as well as the field leadership. But talent-wise, Corey White is very promising, and he's going to be a great guy in the future."
New Orleans signed second-year cornerback Trevin Wade, according to his agent Michael Hoffman. The move won't become official until Tuesday, so the Saints have not yet announced a corresponding move.
Wade, 5-foot-10, 190 pounds, appeared in 13 games with the Cleveland Browns last year, with 17 tackles and one pass defense.