A soft plant fibre spun into thread and cloth.
A familiar fibre with an Arabic ancestor
Arabic
quṭn
‘The fibre’
into Old Italian
Old Italian
cotone
‘The fibre’
into Anglo-French
Anglo-French
cotun
‘The fibre’
into Middle English
Middle English
coton
‘The fibre’
became English
English
cotton
‘The fibre’
Arabic quṭn passed through Old Italian cotone and Anglo-French cotun. Middle English coton precedes today's cotton.
Merriam-Webster also gives the Arabic variant quṭun. It is an alternative form, not another step to put in order.
The noun's Word History gives this sequence and the alternative Arabic form.
The similar middle forms preserve a borrowing route through two languages before English.