Morgan has a torn meniscus and a partial ACL tear. Payton said Morgan was scheduled for surgery Wednesday.
Coleman has a pectoral tear and likely will have surgery Thursday, Payton said.
"Obviously, they are losses, and we've got to find some guys that can step up and take their spot," Payton said.
Coleman had been working with the first team early in camp. His absence means second-year pro Akiem Hicks likely will get a shot at a starting job.
Coleman was entering his 12th season after spending the past two with new Saints defensive coordinator Rob Ryan in Dallas, where he worked as a run-stopping end in Ryan's 3-4 defensive scheme.
"It stinks losing him," said right tackle Zach Strief who has had to block Coleman in practice. "Good player, good teammate, good guy, a lot of knowledge. He's helped me a lot in this camp just understanding how 3-4 guys play. ... Losing any guy hurts but that's the nature of the beast but somebody's got to step up and someone's going to get an opportunity to be successful like he was."
Morgan had seemed to be the leader in the competition for the No. 3 receiver spot.
"You see how hard he has worked this offseason and you see how much he has improved and just how much he's gaining that confidence,'' quarterback Drew Brees said. "His role was there. Then all of a sudden that's taken away."
Information from The Associated Press was used in this report.