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Harpastum

Football was invented about 1600 years after the Roman Empire collapsed so they wouldn't have a word for it.

technically theere wasnt a an ancient game it commenced from 1823. However, the Greek and Roman games of Episkyros and Harpastum respectively have also garnered some interest in tracing the roots of rugby and there are records of a Celtic game called Cad

Around 2000 BC, The Greeks developed a kicking/throwing game, called Episkyros (also known as Phaininda.) It was played primarily by men but women also practiced it. Early balls were made of linen and hair wrapped in string and sewn together. They wouldn't have bounced well. Later Greek models like the Follis were made from inflated pig bladders wrapped tight in leather (pigskin or deerskin.) Another technique used for ball-making was chopping sponges and wrapping them in string and cloth.

The Greek game of Episkyros was later adopted by the Romans, who renamed and transformed it into Harpastum.