The Premier League club's hierarchy are concerned about being dragged into a relegation battle having failed to win a League game in 2023.
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Palace face champions Manchester City, arch-rivals Brighton and league leaders Arsenal in their next three matches.
Vieira was appointed in July 2021 and led the Eagles to a 12th-place finish last term.
But their form has nosedived in recent months and while they are sitting 12th in the table again as it stands, they are only five points clear of the bottom three.
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"If I am sitting in front of you and telling you that I'm not concerned, that would be a lie," Vieira said in his press conference on Friday.
"Of course we are concerned, of course we are aware of our position in the table, but it's important for us to remain calm and keep working and keep improving.
"We have a tough week ahead of us [starting against Man City this weekend] and it will be important to do our best and try to get points. But if we want points we have to play well, so we have to keep doing the right things.
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"At the same time it's important for us to understand what we need to improve to win games and obviously not winning games means we are missing something."
Pushed on whether he thought Palace are in a relegation battle, Vieira chuckled and responded: "You know, the table is what it is.
"We are still five points clear, but anything can happen. We have to know that we are part of the nine or ten teams that have to fight to stat in the Premier League, but we are in a better position than some of the other teams.
"I have been under pressure since the first time I signed for this club. This is what the Premier League is about, every single manager has pressure to perform. I'm no different to any other manager."
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Palace's last victory came against Bournemouth on New Year's Eve.
They have drawn five and lost three of their last eight top-flight games and were also knocked out of the FA Cup with a 2-1 defeat to struggling Southampton at home.
Palace host Manchester City on Saturday and opposite number Pep Guardiola has sympathy for his rival.
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"I've not been with him in the locker room but his experience as a football player [is good], he was in France [as Nice boss] already had and with Palace he's done a really good job."
Palace then travel to Brighton on Wednesday, with a daunting trip to leaders Arsenal, who could well bring an end to their legendary former player's time at Selhurst Park.