Let K be a field and V be a set of rank one valuations of K. The corresponding Tate–Shafarevich group of a K-torus T is Ш(T,V)=ker(H1(K,T)→∏v∈VH1(Kv,T)). We prove that if K=k(X) is the function field of a smooth geometrically integral quasi-projective variety over a field k of characteristic 0 and V is the set of discrete valuations of K associated with prime divisors on X, then for any torus T defined over the base field k, the group Ш(T,V) is finite in the following situations: (1) k is finitely generated and X(k)≠∅; (2) k is a number field.