An angry crowd of around 200 fans gathered outside Celtic Park's main entrance after the shattering play-off round defeat by Maribor, protesting against the club's board of directors, major shareholder Dermot Desmond and the chief executive Peter Lawwell, chanting "sack the board" and "where's the money gone?" and accusing the club of not investing enough in the squad.
The Slovenian champions won 1-0 when Marcos Tavares scored 15 minutes from time to cost Celtic at least £14m in guaranteed Champions League revenue. Instead the Scottish champions will be in Friday's draw for the Europa League groups.
The defeat heaped pressure on Deila, who said he understood the fans' anger. "Of course it's disappointing. This is a feeling I have to take with me. So do the players, the whole staff, the whole club. There are no excuses. This is the level it is. We haven't deserved to be in the Champions League.
"I really understand them [the angry supporters]. The only thing I can do is assure them that we are going to work really hard every day now so that next year we can sit in a better situation and be happy in this moment. We have to be honest. We haven't been good enough. That's the truth.
"The club wants to invest in the team. But I have only been here a short time and it's important we do things in a good way. It is very difficult to get the quality of the players I know the fans want us to get, with the salaries that we have.
"The Europa League is our level now. It is also a very good competition and we are going to get some good games there. We want to be one place - that's the Champions League. Hopefully we can get a new chance next year."
Deila insisted last season's top scorer, Kris Commons, still had a future under him despite not being used in the first leg in Slovenia and only coming on at half-time last night. His introduction added to Celtic's attacking threat but all they needed was a clean sheet and Celtic's poor defending was exposed by Maribor's late pressure.
Deila denied that bringing Commons had been unwise because it weakened the protection of his defence. "I don't think so because if we had continued like in the first half we would have conceded goals. We need to get much more out of what we did in the first half. That wasn't good enough.
"We lifted ourselves in the second half. We had some chances and they scored with their first chance of the second half. We put more pressure on them at the end of the first-half but in the end we didn't lose because of our defending. We lost because we didn't create anything."
Inevitably criticism will intensify on Deila and bookmakers McBookie.com last night issued a media release offering 2/1 that he would be the first manager to be sacked in the SPFL Premiership.
Ante Simundza, the Maribor coach, said he had expected the Celtic players to struggle under the expectations of a big Parkhead crowd, which they did. "It's very hard for every team to play at Celtic Park, and for the home team as well. The fans make a lot of pressure for the home team. The home fans are very tough with the team, they want them always to win."