The Round of 16 draw takes place in Switzerland on Thursday and the Hoops could be set up for a Battle of Britain with the three-time European champions.
"Clubs will look at Celtic as a difficult draw now compared to a few months ago. If you ask Manchester United if they want to come to Celtic Park, with the intensity that goes with it, they'd probably say no," Grant said.
"There would be more of a fear for United because they would be expected to win.
"Celtic are in a good position at this moment, a team people want to avoid.
"If Celtic lose, people will say they've done well to get there, but for teams coming here expected to win it's going to be difficult.
"Teams will know what Celtic Park is like, although seeing from afar is completely different to playing there.
"People will be thinking that if Barcelona can get beaten there then so can they.