Evidence that the motor and the linguistic systems share common syntactic representations would open new perspectives on language evolution. Here, crossing disciplinary boundaries, we explore potential parallels between the structure of simple actions and that of sentences. First, examining Typically Developing (TD) children displacing a bottle with or without knowledge of its weight prior to movement onset, we provide kinematic evidence that the sub-phases of this displacing action (reaching + moving the bottle) manifest a structure akin to linguistic embedded dependencies. Then, using the same motor task, we reveal that children suffering from specific language impairment (SLI), whose core deficit affects syntactic embedding and dependencies, manifest specific structural motor anomalies parallel to their linguistic deficits. In contrast to TD children, SLI children performed the displacing-action as if its sub-phases were juxtaposed rather than embedded. The specificity of SLI's structural motor deficit was confirmed by testing an additional control group: Fragile-X Syndrome patients, whose language capacity, though delayed, comparatively spares embedded dependencies, displayed slower but structurally normal motor performances. By identifying the presence of structural representations and dependency computations in the motor system and by showing their selective deficit in SLI patients, these findings point to a potential motor origin for language syntax.