The tech giant is one of numerous firms hoping to succeed Orange as the company behind the women's literature award, according to The Telegraph.
Kate Mosse, co-founder of the award, previously announced that 18 firms are in contention to become its next sponsor, but the newspaper understands that Apple is the frontrunner.
Apple's interest in fiction is thought to be motivated by the iPad's success in the eBook market. The firm is rumoured to be developing a smaller version of the tablet to compete with leading eReader devices such as the Kindle.
Apple has also made strides as a digital book seller via its iBooks service, though it found itself embroiled in an eBooks price-fixing legal battle with the US Justice Department in June.
Orange has backed the Prize for Fiction since it was founded in 1996, but controversially withdrew its sponsorship in May.