In the absence of the injured Andy Murray, 19-year-old Robson goes into the tournament as Britain's main hope for success despite never having won a main-draw match at Roland Garros.
It is a measure of Robson's talent and liking for the big occasion that her draw against 10th seed Caroline Wozniacki was widely regarded as the pick of the first-round ties.
Wozniacki is on a terrible run having lost four straight matches on clay while Robson has already proved her love of the big occasion by beating Kim Clijsters, Li Na and Petra Kvitova at the last two grand slams.
Courier, a pundit for ITV during the tournament, told Press Association Sport: "She's a very talented player, she's got a lot of natural tennis ability.
"I'm very keen to see how she's playing on the dirt and how she's moving because to do well on clay requires agile feet. I think she could definitely be a dark horse to get to maybe the fourth round."
This year has been patchy for Robson so far but her ability to pull out big results meant she arrived at Roland Garros at a career-high ranking of 35 and on course to be seeded at Wimbledon.
In her last two tournaments in Madrid and Rome, Robson beat world number four Agnieszka Radwanska and Venus Williams, making the losses to lower-ranked players all the more frustrating.
Courier, the winner at Roland Garros in 1991 and 1992, said: "It's great to be able to play well under the spotlight but you also need to learn how to manage yourself when you're playing on the outside courts.
"But we need to be patient with the young players. It's taking longer for them to come through. Robson's very much on the growth curve. She's not at full capacity but hopefully that consistency will come."
Robson used to train in Paris at Patrick Mouratoglou's academy and will take plenty of confidence from her recent results into her first-round match.
She said: "I always enjoy this time of year and, having previously spent a lot of time in Paris, it's nice to be back.
"It is one of my favourite cities and the atmosphere at Roland Garros is always fun. I feel that I am in a good place and playing good tennis so I am looking forward to getting on the match court."
Robson goes into the French Open without a full-time coach after parting ways with Zeljko Krajan earlier this month.
The Londoner has been working with Sven Groeneveld through her clothing sponsor adidas but will not be able to have the Swede in her corner for her first-round match as he also works with Wozniacki.
Krajan was regarded as a hard taskmaster, which did not fit with Robson's more relaxed attitude, and the teenager is taking her time in finding his successor.
Courier said: "It's such a close relationship. You spend so much time on the road together so it needs to be right.
"From the outside, what I understand is sometimes the information (from Krajan) was fairly negative, and if that's not balanced then it's not going to work.
"It's important to find someone who fits into the psyche of the player. She needs someone now who can help her get to the next level."
Robson is one of three British players in the women's draw along with Heather Watson and Elena Baltacha, who are both grateful to be in Paris.
Watson, who meets Swiss Stefanie Voegele, returns after two months out with glandular fever while Baltacha, up against New Zealand's Marina Erakovic, is playing her first grand slam after missing eight months following ankle surgery.
There is no doubt about the favourite for the women's title. World number one Serena Williams is on a career-best winning streak of 24 and has lost just three times in her last 70 matches.
That run began a year ago after a shattering first-round loss at Roland Garros to Frenchwoman Virginie Razzano, but Courier does not expect there to be any lurking demons.
He said: "I don't think Serena has a real challenger other than herself. But this year is very different. Her loss last year in the first round gave her a new outlook.
"It was a real wake-up call. Players respond one way or another. I think she took it as an affront to her talent.
"Her result last year shouldn't be weighing on her mind. I think she deserves to feel like the clear favourite and, for me, she wins going away."