In my accent, at least, "bicycle" is pronounced /bɑɪsɪkʊl/, but it's shortened to /bɑɪk/, and not /bɑɪs/. The latter would be analogous to how some people shorten "decent" to "dece" /diːs/, but it doesn't happen for bike. Is there a historical reason why this is so?
The OED reports that the word was first attested in Cyclist & Bicycling & Tricycling Trades Rev. 29 Dec. 100/1, "We can conscientiously recommend it as an excellent shillingsworth of ‘bike’ and ‘trike’ literature". "Bicycle" was borrowed much earlier from French. There not being much evidence about the history of the word(s), one can only speculate. There is some resemblance to hypocoristic reduction and mutilation process for creating affectionate versions of words, like Art ← Arthur, Ginnie ← Virginia, Bob ← Robert. The scare quotes in the citation probably indicates a judgment that the form is seen as "cute". There is certainly nothing regular about the process. It might have been a writer's cute reintroduction of the velarity of original Greek κ.