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The former Hoops defender admitted he doesn't think O'Neill knows his best starting XI yet and pointed to several key positions where he feels uncertainty remains and reckons Viljami Sinisalo doesn't make enough saves at European level.
Mulgrew also reckons big-money striker Kasper Hogh continues to lead the line largely because of the investment made in him, despite an obvious lack of confidence.
Speaking to Premier Sports Ireland, he said: "I don't think he does (know his best team). Hogh has been bought for a lot of money, that is why he is leading the line. He's clearly lacking confidence.
"You can understand why Martin O'Neill continues to keep playing him. He wants to get his main striker, a lot of money has been spent on him, up in confidence. Hopefully he can get a goal that lifts him."
However, Mulgrew insists Celtic's strongest attacking setup is becoming increasingly obvious.
He added: "I think it is as clear as day that you can see now that Hassan on the right and Duran up top is the strongest way to go."
The ex-Celtic defender also highlighted the absence of Liam Scales, suggesting the Republic of Ireland international's importance is often underestimated.
Mulgrew was critical of Auston Trusty for the goal conceded before half-time and reckons Scales would have prevented it.
He added: "Scales, you can only assume he's got a knock and is being rested. Because I think you see how big a player he is for Celtic. It's not always appreciated but he is somebody who heads the ball in the box and defends well."
"The Trusty one, I think he should have blocked the goal just before half-time. I think Scales blocks that, gets his leg in and stops the line of the ball."
But one of his biggest worries centred around who is occupying the nets after the goals rained down once more.
He said "And Sinisalo, for me, you're still scratching around for a No.1. Johnstone has come in and he's had a bad time at Ibrox and came out the team, with Sinisalo back in. Continually, for me, in Europe, Sinisalo doesn't make the necessary saves."