There is a general consensus that many children and adults with dyslexia and/or specific language impairment display deficits in auditory processing. However, how these deficits are related to developmental disorders of language is uncertain, and at least four categories of model have been proposed: single distal cause models, risk factor models, association models, and consequence models. This study used children with mild to moderate sensorineural hearing loss (MMHL) to investigate the link between auditory processing deficits and language disorders. We examined the auditory processing and language skills of 46, 8-16year-old children with MMHL and 44 age-matched typically developing controls. Auditory processing abilities were assessed using child-friendly psychophysical techniques in order to obtain discrimination thresholds. Stimuli incorporated three different timescales (µs, ms, s) and three different levels of complexity (simple nonspeech tones, complex nonspeech sounds, speech sounds), and tasks required discrimination of frequency or amplitude cues. Language abilities were assessed using a battery of standardised assessments of phonological processing, reading, vocabulary, and grammar. We found evidence that three different auditory processing abilities showed different relationships with language: Deficits in a general auditory processing component were necessary but not sufficient for language difficulties, and were consistent with a risk factor model; Deficits in slow-rate amplitude modulation (envelope) detection were sufficient but not necessary for language difficulties, and were consistent with either a single distal cause or a consequence model; And deficits in the discrimination of a single speech contrast (/bɑ/ vs /dɑ/) were neither necessary nor sufficient for language difficulties, and were consistent with an association model. Our findings suggest that different auditory processing deficits may constitute distinct and independent routes to the development of language difficulties in children.