Key points How parallel are the primate visual pathways? In the present study, we demonstrate that parallel visual pathways in the dorsal lateral geniculate nucleus (LGN) show distinct patterns of interaction with rhythmic activity in the primary visual cortex (V1). In the V1 of anaesthetized marmosets, the EEG frequency spectrum undergoes transient changes that are characterized by fluctuations in delta‐band EEG power. We show that, on multisecond timescales, spiking activity in an evolutionary primitive (koniocellular) LGN pathway is specifically linked to these slow EEG spectrum changes. By contrast, on subsecond (delta frequency) timescales, cortical oscillations can entrain spiking activity throughout the entire LGN. Our results are consistent with the hypothesis that, in waking animals, the koniocellular pathway selectively participates in brain circuits controlling vigilance and attention. Abstract The major afferent cortical pathway in the visual system passes through the dorsal lateral geniculate nucleus (LGN), where nerve signals originating in the eye can first interact with brain circuits regulating visual processing, vigilance and attention. In the present study, we investigated how ongoing and visually driven activity in magnocellular (M), parvocellular (P) and koniocellular (K) layers of the LGN are related to cortical state. We recorded extracellular spiking activity in the LGN simultaneously with local field potentials (LFP) in primary visual cortex, in sufentanil‐anaesthetized marmoset monkeys. We found that asynchronous cortical states (marked by low power in delta‐band LFPs) are linked to high spike rates in K cells (but not P cells or M cells), on multisecond timescales. Cortical asynchrony precedes the increases in K cell spike rates by 1–3 s, implying causality. At subsecond timescales, the spiking activity in many cells of all (M, P and K) classes is phase‐locked to delta waves in the cortical LFP, and more cells are phase‐locked during synchronous cortical states than during asynchronous cortical states. The switch from low‐to‐high spike rates in K cells does not degrade their visual signalling capacity. By contrast, during asynchronous cortical states, the fidelity of visual signals transmitted by K cells is improved, probably because K cell responses become less rectified. Overall, the data show that slow fluctuations in cortical state are selectively linked to K pathway spiking activity, whereas delta‐frequency cortical oscillations entrain spiking activity throughout the entire LGN, in anaesthetized marmosets.