The 20-year-old failed to make a first-team appearance for Spurs, spending most of his time on their books away on loan at Luton Town, Rangers, Colchester United and Peterborough United.
"This is a great day for our club," Olimpija director Ranko Stojic said. "Nathan offers both speed and technique that we will support in the right direction to develop its huge potential."
Oduwa represented England at Under-17, Under-18 and Under-20 level before featuring for Nigeria Under-23s at the 2016 Olympic Games in Rio de Janeiro.
Olimpija are currently second in the Slovenian first division, having won the league for the first time last season, and were able to complete the deal due to the Slovenian transfer window remaining open until February 15.