Entente support for Italian claims in Libya goes back farther than Italy's entry into the Triple Alliance in 1882, beginning during the French occupation of Tunisia in 1881 alongside Britain's occupation of Cyprus the same year, the same event that would spur Italy's entry into the Triple Alliance.

Italy made a claim to Tunisia shortly after unification in the 1860's to serve as the first springboard into entering the Scramble for Africa that would ensue throughout the late 1800's, but France and Britain occupied their respective territories in the Mediterranean before Italy. This infuriated Italian diplomats and shifted public opinion against France, serving as the cause for Italy's entry into the Triple Alliance with Germany and Austria-Hungary the next year.

But this doesn't mean that Italian diplomats counted this as a loss and refused to cooperate with France and Britain. When Italian delegates expressed their opposition to the French seizure of Tunisia, French diplomats replied that Tripoli would serve as a 'reimbursment' for Italian claims on Tunisia.

In 1902, this agreement only exchanged in notes was solidified in a secret treaty signed between France and Italy, which France agreed to no intervention in Tripoli and Morocco, leaving Libya slated for Italian annexation. Great Britain also made a statment regarding the issue that same year, stating: "any alteration in the status of Libya would be in conformity with Italian interests.". The issue of Morroco in that treaty would see a different fate, with France and Spain dividing the territory a decade later after the Moroccan Crises agitated by the German Empire's demands for territorial compensation after Morroco was made a French protectorate.

Italy would not fully invoke its territorial claim on Libya until the summer of 1911. reacting to the Agadir Crisis in Morroco and with the Army's confidence that campaign in Libya would be successful, Italy presented an ultimatum to the Ottomans for control of Libya.

Germany and Austria-Hungary, Italy's partners in the Triple Alliance advised for a mediated agreement between the Italians and Turks. Their reasons for this was to keep good relations with the Ottoman Empire and secure them as an additional ally against the Entente. A compromise was suggested by the Ottoman delegation to transfer control of Libya to Italy in all practical manners while still maintaining Libya's status as a tributary to the Ottoman Empire, the same agreement Britain and the Ottomans made concerning Egypt decades prior. Italy rejected this offer and declared war, annexing Libya and some Aegean islands (most important of which was Rhodes) once the Treaty of Ouchi was signed the next year.

Italy acted in a swift and aggressive manner because they believed the opening for their claim on Libya was closing due to the Agadir Crisis in Morroco, where the country became a French protectorate despite the promise of non-intervention in the country in the 1902 secret treaty. To the Italians, if action wasn't taken soon, then their claim to Libya could be ignored just as it was in Tunisia.

The Entente Powers upheld the 1902 treaty and expressed their support for Italian annexation of their claim in Libya, but their reasoning for this was twofold. One was the simple fact that they honored their agreements with Italy, the second was to jockey favor with the Italian government and the public against the Triple Alliance. The Entente's support for Italy's claims and the Triple Alliance's call for mediation was very fresh in the minds of the Italians once World War I broke out, and ultimately contributed to Italy's withdraw from the Triple Alliance and entering the war on the side of the Entente upon the signing of the Treaty of London in 1915.

source:

Italo-Turkish Diplomacy and the War Over Libya: 1911-1912 by Timothy Winston Childs

Edit: typo