I think this is a question that I can actually answer. I wrote my undergraduate thesis on the French Foreign Legion's demographics from 1914 to 2014.

The French Foreign Legion was very fractured during the Second World War. There was no central command that organized all of the different garrisons and formations of the Legion. You have Legionnaires fighting for the allies, for the axis and not fighting at all. Furthermore, they change sides depending upon France's status as a liberated/occupied state.

A large part of the Legion fell under the command of the Vichy Government, who subsequently released 2,000 German Legionnaires to the Nazis. Most of them were communists, or of Jewish heritage. On a related note, when Italy joined the war, many of the Italian Legionnaires were sent back to their homeland. To fill the manpower shortages France ends up activating many elderly, unfit Legionnaires living in retirement. I should also note that not all German legionnaires were sent back to Germany. Germans made up an incredibly large part of the Legion's non-commissioned officers corps. In 1934, 44% of the Legion was German/German speaking and 21% of the NCOs were German/German speaking. This prevalence of German speakers actually alarmed the French government, who worried that if another war broke out that the Legion would be incapable of defending France from a German invasion. There were actual fears that the Germans were purposely sending young men to the Legion in order to subvert it. There is little evidence that this was the case, but the paranoia was sufficient enough that little in the way of material/emphasis was placed on the Legion in the years leading up to World War 2, and by 1939 it was a bent and broken sword that France would have much rather pretended did not exist.

So, you have some Legionnaires fighting for Vichy France. Those that were stationed in North Africa get folded into the German Afrika Korps. They fight in North Africa, and they fight in the Levant. When the Germans are defeated on that continent, many of the Legionnaires switch sides and begin fighting for the allies again.

In Asia, you have the French garrisons in Indochina. Eighty to one hundred Legionnaires escape European France and seek solace with the 5th Infantry Regiment in Indochina. The French forces in this part of the world sit out the war. They allow the Japanese to occupy Indochina, but are not taken into custody by Imperial forces.

In 1941, you have Legionnaires landing with the allies during Operation Torch (30% of them were Spaniards who had fled Spain when they lost the Civil War in that country). They had fought in Norway, and were under command of the Armee d'Afrique, which would go on to see service not only in North Africa, but Italy, France and even Germany. This is one of the fighting formations of the exiled French Government under Charles de Gaulle.

Now, I must admit that I do not know what you mean by, "...any attempts by Petain's government to establish a foreign legion from Axis powers." The best I can gleam off of that portion of your question is whether or not Petain's government accepted foreign volunteers into it's armed forces. I must admit, I am ignorant, but an educated guess based upon my research would suggest that the Legion under Petain was struggling to survive as a military unit; fending off the awkward attempts of the other axis powers to absorb it completely.

Sources:

Geraghty, Tony. March or Die: France and the Foreign Legion. London: Grafton, 1986. Print.

McLeave, Hugh. The Damned Die Hard. New York: Saturday Review, 1973. Print.

Porch, Douglas. The French Foreign Legion: A Complete History of the Legendary Fighting Force. New York, NY: Harper Collins, 1991. Print.