The match is scheduled for Tuesday, although Monday's play was cancelled because of the rain and there is more wet weather predicted.
Second-seed Murray has beaten Gasquet in all of their Grand Slam meetings.
"I'm pumped to be in the quarters of a slam. The atmosphere will be tough but I don't mind that," said Murray, 29.
"I've played a number of times against French players here in difficult atmospheres and I managed OK, so I'm sure I'll be fine."
Murray has beaten 29-year-old Gasquet four times in Grand Slam matches - twice at Roland Garros, coming from two sets down in a first-round match in 2010 and one set behind in their last-16 tie in 2012, and twice at Wimbledon.
Gasquet defeated fifth seed Kei Nishikori on Sunday to reach the quarter-finals of the French Open for the first time in his career.
"To beat one the best players in the world like that was a great feeling for me," Gasquet, ranked 12th in the world, told BBC Radio 5 live.
"The most important tournament for me is here in Paris, on the Philippe Chatrier Court with the crowd cheering for me, like they did.
"I was very close to beating Andy in 2008 at Wimbledon when I was serving for the match, but he managed to come back.
"He is number two in the world, and will be clearly the favourite. But I have nothing to lose, and I will do my best to win."
