The 42-year-old took charge on an interim basis last month following the sacking of Unai Emery but has been unable to oversee a turnaround in fortunes with the north London outfit.
A 3-0 home defeat to the reigning Premier League champions means the Gunners have won just one of their last 12 games in all competitions.
Ljungberg's position remains unclear as the club continue to plan for a full-time successor to Emery, with the likes of Carlo Ancelotti, Mikel Arteta and Patrick Vieira in the running alongside the former Sweden international.
With the club now as close to the relegation zone as they are to the top four, Ljungberg feels it would be in everyone's best interest if an appointment were made sooner rather than later after he was also told he could not recruit his own backroom staff before a decision is taken.
"I think, as I've said to the club, it's a great, great honour to do this," he said. "Of course, Per (Mertesacker) is the academy manager and he's doing two jobs in one go. I think it needs clearing up to make a decision so that everybody knows.
"That's something I've said. It's totally up to the club. I'm very honoured and trying to do things as well as I can, but I think that it would be good to make a decision regardless of what it is."
Arsenal were jeered off by their own fans at half-time but, by the end of the game, the supporters were singing their support for the club as anger appeared to turn to apathy.
"Of course, I'm an Arsenal man and we try everything," added Ljungberg. "But at the same time we need to be a little bit realistic. There is a gulf in quality between City, Liverpool and the rest. They showed that."
The Gunners are back in action on Saturday when they make the trip to Goodison Park to face a resurgent Everton side who have beaten Chelsea and drawn with Manchester United under their interim boss Duncan Ferguson.