The defender knows the Parkhead club's decision to allow midfielder Victor Wanyama, striker Gary Hooper and defender Kelvin Wilson to leave will come under scrutiny should they fail to overturn a 2-0 deficit in the Champions League play-off tie.
Manager Neil Lennon has brought in Amido Balde, Virgil van Dijk, Derk Boerrigter and Steven Mouyokolo, but the Hoops manager continues to search for more players to bolster his squad although no new signing would be eligible for the second leg in Glasgow next Wednesday.
"We are not using that as an excuse," said Mulgrew. "We knew these questions would come but we have a lot of belief in each other, we have a big squad and enough players to deal with the challenge.
"You are hoping you get a more positive result but it could have been worse.
"We have to look at the positives and we have a real chance if we approach it positively next week.
"We are not thinking along the lines (of non-progression).
"We knew this tie was over two legs and we are looking to next week to turn it around."
Mulgrew is hoping the crowd will play their part in the Hoops' rescue mission.
The Scotland international said: "The crowd will get right behind us and we will really need to go for them.
"We we were well prepared, we knew what was coming and we know what we will get again next week but we have to stand up to it better and score a few goals."