The winger damaged his medial knee ligaments during the pool stages defeat to Northampton last month and, while he is targeting an April return, he confirmed that he probably won't be rushed back in time for the last-eight meeting.
The Northerners last played at English rugby's HQ when losing the Heineken Cup final to Leinster in May. Saracens had been expected to name their new stadium, Allianz Park, as the venue, but opted for the bigger stadium for the April 6 clash.
The news that Bowe is likely to miss the quarter-final means that his chances of making the Lions tour have receded, although coach Warren Gatland knows all about the winger's talents after working with him in South Africa in 2009, where the 28-year-old started all three Tests.
He is due to name his squad between the European quarter and semi-finals.
Having undergone surgery six weeks ago, Bowe has been able to apply pressure to the joint in recent days as he works his way towards a return.
And he revealed that the injury, which occurred in the closing stages of the defeat to the Saints that ultimately led to Ulster having to go away for their quarter-final, was progressing nicely.
"I'd love to say I'll be back for the quarter-final, but I think that could be a week or two too soon," he said after collecting his scholarship to Hibernia College.
"I'd love Ulster to win it and be involved in all competitions. We have put ourselves in a strong position and hopefully we can keep going the way we have been.
"I have been lucky with the injury. To not catch my cruciate was lucky, and there is a nerve attached to the fibia bone and not catching that was a blessing, because that could have been curtains altogether.
"Three to four months is nothing really."
Meanwhile, Bowe is confident that young wingers Simon Zebo and Craig Gilroy can make an impact in his absence as Ireland take on Wales on Saturday.
"It is going to be a massive challenge for Craig and Zebs coming in. It is going to be exciting for Irish rugby; we have an exciting back three this weekend," he said.
"It is hard to watch. I want to be there, fighting for my place. It is different from not being selected and I have to look at it with a positive outlook and support the guys."