Let P be the set of all prime numbers, I be a set and σ={σ_i∣i∈I} be a partition of P. A finite group is said to be σ-primary if it is a σ_i-group for some i∈I, and we say that a finite group is σ-solvable if all its chief factors are σ-primary. A subgroup H of a finite group G is said to be σ-subnormal in G if there is a chain H=H₀≤H₁≤⋯≤H_n=G of subgroups of G such that H_i-1 is normal in H_i or H_i/(Hi-1)H_i is σ-primary for all 1≤i≤n. Given subgroups H and A of a σ-solvable finite group G, we prove two criteria for H to be σ-subnormal in ⟨H,A⟩. Our criteria extend classical subnormality criteria of Fumagalli [5], which themselves generalize a classical subnormality criterion of Wielandt [13].