Let {ϕn(x), n = 1, 2,...} be an arbitrary complete orthonormal system on the interval I:= [0, 1]which consists of a.e. bounded functions. Suppose that E0 ⊂ I2 is any Lebesgue measurable set such that μ2E0 > 0, and φ, φ(0) = 0, is an increasing continuous function on [0, ∞) with φ(u) = o(u ln u) as u → ∞. Then there exist a function f ∈ L1(I2) and a set E0′, ⊂ E0, μ2E0′ > 0, such that $$\int_{I^2 } {\phi (|f(x,y)|)dxdy < \infty } $$ and the sequence of double Cesaro means of Fourier series of f with respect to the system {ϕn(x)ϕm(y): n,m = 1, 2,...} is unbounded in the sense of Pringsheim (by rectangles) on E0′. This statement gives critical integrability conditions for the Cesaro summability a.e. of Fourier series in the class of all complete orthonormal systems of the type {ϕ n(x)ϕm(y): n,m = 1, 2,...}.