De Gea watched from the stands as United began their season with a 1-0 win over Tottenham on Saturday.
Manager Louis van Gaal selected Sergio Romero ahead of De Gea as he felt the 24-year-old would not have been able to focus properly on the match after a summer of speculation regarding his future.
Real Madrid are interested in signing the Spain international, but United are refusing to do business with the 10-time European champions unless they get defender Sergio Ramos in return.
It is understood that United are still hopeful of keeping De Gea at United beyond September 1 even though the player was left out of the match day squad and may not feature for the club until after the transfer window closes.
Schmeichel admits the situation is far from ideal and should have been resolved before the season began.
“If the player is not happy; his agent, himself and the club that want him should have sorted this a long time ago,” the former United goalkeeper told talkSPORT.
“It’s sad that it’s dragged into the season.”
After a series of excellent displays, De Gea won United’s player of the year award for a second successive season in May.
The former Atletico Madrid stopper was also named in the Professional Footballers’ Association team of the year.
Schmeichel wants De Gea to snub a move to the Spanish capital, but concedes there is little United can do if the player wants to leave.
“I’d really like De Gea to stay,” said Schmeichel, who won the Premier League five times during his eight-year spell at Old Trafford.
“Two years running he’s been player of the year. You don’t let players like that go but if he wants to go, let him go and let Manchester United move on.”
Schmeichel is surprised United have lined up Cillessen as a replacement for De Gea.
Van Gaal substituted the Ajax goalkeeper for Tim Krul in the dying minutes of extra-time during Holland’s win over Costa Rica at the World Cup last year because he felt the Newcastle man was better at saving penalties.
Cillessen did not respond well, kicking a water bottle as he walked off the pitch, but the tactic worked as Krul saved two spot-kicks to send Holland through to the semi-finals.
Schmeichel thinks the substitution showed Van Gaal does not have complete faith in Cillessen and he believes Tottenham goalkeeper Lloris would be a better option for United if De Gea leaves.
“Cillessen is a good goalkeeper, but (Van Gaal) took him off for the penalty shoot-out at the World Cup and that shows some kind of no trust,” Schmeichel said.
“I think he doesn’t trust him 100 per cent and I wouldn’t like a goalkeeper like that at Manchester United.
“If De Gea goes I would love Lloris to come to Manchester United. He has exactly the style of play and he is proven in the Premier League.”
United’s back-up goalkeeper Sam Johnstone, meanwhile, says his future has been left in limbo because of the uncertainty surrounding De Gea’s future.
The 22-year-old was set to rejoin home town club Preston, where he was on loan last season, but De Gea’s apparent inability to focus properly, and the fact that Victor Valdes has been frozen out of the first-team equation, means he has become Van Gaal’s second-choice goalkeeper behind Romero.
The goalkeeper has not given up hope of securing a loan move back to the Championship side, though.
“I’m not sure. Nothing’s been said really,” said Johnstone, who was on the bench for Saturday’s win.
“We’re waiting on different situations and have to see how it pans out.
“I need to play games but on the other hand I am contracted to Manchester United so I have to see what happens here first.
“If I wasn’t getting the game time here then I could go on loan later in the season. That’s another possibility so we will have to see what happens.”
De Gea was later included in United’s Champions League squad for the two-legged play-off against Club Brugge.
However, even if he appeared in either of the matches – which take place on August 18 and August 26 – he would still be eligible to represent another club in the group stages of the competition this season, according to UEFA rules.
Also named in the 28-man group were all five of United’s summer signings – Sergio Romero, Matteo Darmian, Morgan Schneiderlin, Bastian Schweinsteiger and Memphis Depay.
Man Utd squad: David de Gea, Anders Lindegaard, Sergio Romero, Sam Johnstone; Phil Jones, Marcos Rojo, Jonny Evans, Chris Smalling, Luke Shaw, Paddy McNair, Matteo Darmian, Tyler Blackett; Adnan Januzaj, Michael Carrick, Daley Blind, Ashley Young, Ander Herrera, Antonio Valencia, Marouane Fellaini, Morgan Schneiderlin, Bastian Schweinsteiger, Jesse Lingard, Andreas Pereira; Memphis Depay, Juan Mata, Wayne Rooney, Javier Hernandez, James Wilson.