The duo saw red within three minutes of each other in Thursday's fixture at Newcastle United, allowing the home side to complete a comfortable 5-1 triumph.
Hughes will now be without the pair for Sunday's trip to White Hart Lane to face Tottenham.
Goalkeeper Asmir Begovic will also miss out with a broken finger, while centre-back Robert Huth is not expected to recover in time from the knee injury that has sidelined him for the past month.
"We will be without both of them (Wilson and Whelan)," Hughes told the club's official website. "It's not ideal but that is the way it is - and we have to get on with it.
"They will miss this game, as will Asmir, who looks likely to be out for between four and six weeks.
"The match will more than likely come too soon for Robert as well, so we will have to make a couple of changes to the team."
Despite his squad being depleted for the match against Tim Sherwood's outfit, Hughes is confident of challenging Tottenham and has welcomed the opportunity to respond quickly to the Newcastle defeat.
He added: "We are exasperated by the events of Boxing Day, but we have another game coming our way very quickly which I think is a good thing for us.
"Obviously we will be without a few players now, but we still have players who have the quality to go to Tottenham and cause them a lot of problems."
