At any stage of their lifecycle, mRNAs are coated by specialized proteins. One of few circumstances when free mRNA appears in the cytosol is the disassembly of polysomes during the stress-induced shutdown of protein synthesis. Using quantitative mass spectrometry, we sought to identify the free RNA-interacting cellular machinery in heat-shocked mammalian cells. Free RNA-associated proteins displayed higher disorder and larger size, which supports the role of multivalent interactions during the initial phase of the association with RNAs during stress. Structural features of the free RNA interactors defined them as a subset of RNA-binding proteins. The interaction between these assembled proteins in vivo required RNA. Reconstitution of the association process in vitro indicated a multimolecular basis for increased binding to RNA upon heat shock in the cytosol. Our study represents a step toward understanding how free RNA is processed in the cytosol during proteostasis stress.