The Albion are hoping they can convince Roberto De Zerbi to stay for another season.
However, the Italian is wanted by several top teams, including Liverpool and Bayern Munich.
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Speaking to TNT Sports, Barber said: Every time that you do well in the Premier League, expectations go up, not just from your fans but also from your coaches, players and staff.
It's a challenge to make sure you can meet those expectations but it's something we intend to try.
As always in football there's no guarantee. With Roberto, it doesn't matter if we're sixth; he'd want to be fifth. If we were fifth, he'd want to be fourth, and so on.
That's the way that he's built, he's commanding, but we like it that way. He pushes us all to be better."