The meteorite, which was discovered in the Moroccan west Sahara Desert, is known as Oued Awlitis 001 and will cost the museum €110,000.
However, the museum does not have an aquisition budget so is turning to Crowdfunding to try to purchase this piece of the moon with the help of the public.
Members of the public can buy a piece of the moon for as little as €5 but larger donations will result in the donor's name making it onto the plaque to be displayed with the exhibit.
The museum says that the meteorite will assist in learning more about the moon and the unknown space around us.
Meteorite researcher and museum curator Ludovic Ferrière explained that the purchase of the rock was of great importance.
"It will help us to understand so much more about the moon which is important not only for science but also for generations to come. It would be wasted in a private collection so we want to make it available to the public through this Crowdfunding project."
He also pointed out that the moon missions so far have only explored a very small portion of our moon and this meteorite could produce important results for researchers.
The museum already has an impressive meteorite collection which is the biggest and oldest in the world with 1,100 pieces currently on display.
The Oued Awlitis 001 is on display at the museum until the end of the year as a kind of advertisement to encourage visitors to help purchase the meteorite.
So far, the crowdfunding initiative has reached two percent of its goal, with €3,270 raised towards the purchase of the rock.
Ferrière added: “Anyone who decides to make a sizable donation to the purchase of the meteorite could even get the chance to hold it. Though only with gloves on.”