Murray, who has never been beyond the last-four at Roland Garros, admitted to feeling some tiredness after playing for a third straight day, before seeing off No24 seed Verdasco 6-4, 7-5, 7-6.
A semi-final against “King of Clay” Rafael Nadal, winner of eight of the last nine French Opens, beckons, but Murray must first overcome Monfils, who has enjoyed a renaissance in the last 12 months, battling back from outside the world’s top 100 to be seeded 23rd and is relishing their first clash in four years.
“He’s one of the best players to watch, he’s so entertaining and a really nice guy,” said Murray, the reigning Wimbledon champion. “I’m really looking forward to it.
“I was a bit nervous at the end [against Verdasco], but I’m glad I managed to get through. I was trying to play more aggressive because I knew I’d be a little bit tired in my legs.
“Thankfully the balls were going in, that doesn’t always happen.”
Nadal dropped just four games in victory over Dusan Lajovic and will meet last year’s runner-up David Ferrer in their quarter-final, also set to commence tomorrow.
World No2 Novak Djokovic is set to take on Canada’s Milos Raonic today in the last-eight, while No6 seed Tomas Berdych meets Latvia’s Ernests Gulbis.
In the women’s singles draw, No4 seed Simona Halep beat America’s Sloane Stephens in straight sets, while Italy’s Sara Errani got the better of No6 seed Jelena Jankovic.
Bob and Mike Bryan, 15-time men’s doubles grand slam champions, lost their quarter-final in straight sets to Spanish pair Marcel Granollers and Marc Lopez, while third seeds Daniel Nestor and Nenad Zimonjic also tumbled out.