  Enriched motivic $\mathcal A$-spaces are introduced and studied in this
paper, where $\mathcal A$ is an additive category of correspondences. They are
linear counterparts of motivic $\Gamma$-spaces. It is shown that rational
special enriched motivic $\widetilde{\mathrm{Cor}}$-spaces recover connective
motivic bispectra with rational coefficients, where $\widetilde{\mathrm{Cor}}$
is the category of Milnor--Witt correspondences.
