One explanation for the form with /k/ is that bicycle was parsed to bi(cy)c(le). An alternative explanation is that it was parsed to bic(ycle) but since speakers are aware of a general /k/~/s/ alternation (as in electric ~ electricity etc.), the softened /s/ was restored to a default /k/ when the “ending” -ycle was dropped.[1] Similar cases are merc /mɜɹk/, spec /spɛk/ for mercenary, specify. It seems unlikely, however, that this process is purely phonological and not at least partially based on the spelling ⟨c⟩.
Bike (n.)
Etymology
Clipping of bicycle. First attested in 1882.
One explanation for the form with /k/ is that bicycle was parsed to bi(cy)c(le). An alternative explanation is that it was parsed to bic(ycle) but since speakers are aware of a general /k/~/s/ alternation (as in electric ~ electricity etc.), the softened /s/ was restored to a default /k/ when the “ending” -ycle was dropped.[1] Similar cases are merc /mɜɹk/, spec /spɛk/ for mercenary, specify. It seems unlikely, however, that this process is purely phonological and not at least partially based on the spelling ⟨c⟩.
wiktionary
I’d add that Wiktionary itself cites this 1976 article by Robert Hausmann.