Interesting question, I have not heard of this until today but upon some research in Turkish media I can confirm some of the highlights in this claim. An news article from CNN Türk (2015) claims that in the 1860's the Ottoman Sultan Abdülaziz commissioned a monument, that were supposed to be like the statue of liberty and were to made the same sculpturer Frederic Auguste Bartholdi that lter would design the statue of liberty. It were supposed to be set up by the Suez Canal at Port Said, but officials in Egypt were not found of the idea so they canceled. But the sultan had already paid some money upfront, and the desing were to be women figure in wails. So the tale goes and when Bartholdi were to design the statue of liberty he kinda reused his design for the statue that were comissioned by the Ottomans.[1]

After digging through internet archives I could not find valuable or useful source about this, but then I found the Neo-Ottoman Islamist and Pseudo Historian Mustafa Armağan had written a article about this and started the hoax. And this man is widely criticized for faking sources and making up things by respected Turkish historians in the field of Ottoman (like Ilber Ortaylı and Halil İnalcik). Mustafa Armağan is generally known for producing some paparazzi like history articles on far-right islamist media, without citing reliable sources or even having a touch of source criticism in mind. In his news articles he commonly attacks secular views in history, and puplishes made up stories from the late Ottoman period to glorify the Ottoman Empire and pan-islamic nationalism. The fact that he did not know how little influence the Ottoman Sultan had in Egypt, at this time Egyptian Province were autonomous and were heavily influenced by foreign intervention. And out no reason would want plant a great statue on the shores of canal build by foreigners, in a time where the Egyptian Khedive was trying to get out Ottoman sphere of influence. The claim and the correlation between the historical context is just not there to found. [2]

Having no sorces than internet articles that pinpoint back to a known fake/propagandist historian, I can assure you that the Ottomans or the Turks did not pay for the Statue of Liberty. Maybe the sultan had commissioned a statue, but seriously I searched for sources in Turkish and English and could not find anything that seemed to be academic or historical source material that pi n pointed the Ottomans paying for the statue being made. From this I would say this definitely is a hoax until someone can find some sources to support this theory.

[1]

https://www.google.dk/amp/www.cnnturk.com/amp/yasam/ozgurluk-heykeli-aslinda-osmanlinin-mi

[2]

http://www.zaman.com.tr/mustafa-armagan/ozgurluk-heykelini-abdulaziz-mi-yaptirdi_1044109.html

Sayyid-Marsot, Afaf Lutfi. A short history of modern Egypt. Cambridge University Press, 1985.

Hunter, F. Robert. Egypt under the khedives, 1805-1879: from household government to modern bureaucracy. American Univ in Cairo Press, 1999.

Mitchell, Timothy. Colonising Egypt: With a new preface. Univ of California Press, 1991.

Shaw, Stanford J., and Ezel Kural Shaw. History of the Ottoman Empire and Modern Turkey: Volume 2, Reform, Revolution, and Republic: The Rise of Modern Turkey 1808-1975. Vol. 11. Cambridge University Press, 1977.

Here is some of Armağans articles, its basically useless as historical material. No sources, no real argumentation and just filled I knows and slurs. And all from his very own homepage. And if you know Turkish, or just take a look at some of these writings you can see how he has pattern of propaganda and only looks from his own perspective.

An article about how the british tricked the Sultan to hand over Cyprus http://www.mustafaarmagan.com.tr/kibrisi-ingiltereye-sultan-abdulhamid-mi-verdi/

The Ottomans found the cure for smallpox, but the british protested against it http://www.mustafaarmagan.com.tr/cicek-asisini-osmanlilar-buldu-ingilizler-protesto-etti/

President Jefferson had read the Quran http://www.mustafaarmagan.com.tr/abdnin-efsane-baskani-jefferson-kuran-okumus/