In the case when a sequence of d-dimensional vectors nk = (nk1, nk2, ..., nkd) with nonnegative integer coordinates satisfies the condition $$ n_k^j = \alpha _j m_k + O(1),k \in \mathbb{N},1 \leqslant j \leqslant d, $$ where α1,..., αd > 0, mk ∈ ℕ, and limk→∞mk = ∞, under some conditions on the function ϕ: [0,+∞) → [0,+∞), it is proved that, if the trigonometric Fourier series of any function from ϕ(L)([−π, π)) converges almost everywhere, then, for any d ∈ ℕ and all f ∈ ϕ(L)(ln+L)d−1([−π,π)d), the sequence Snk (f, x) of the rectangular partial sums of the multiple trigonometric Fourier series of the function f, as well as the corresponding sequences of partial sums of all of its conjugate series, converges almost everywhere.