Cufre picked up a leg injury in a clash with his friend Cafu - but fears that he had broke something proved unfounded.
However, a deep wound required medical attention and the player faces a spell out of the side.
Cufre is unlikely to be fit in time for the match against Udinese on April 3 - which causes a big problem for coach Bruno Conti.
Conti is already without the banned pairing of Francesco Totti and Christian Panucci who were sent off in the 2-0 defeat to Milan - and now looks to have lost a third key man.