to introduce (an idea) or suggest (something unpleasant) in a subtle or oblique manner

My liveliness and your solidity would produce perfection. – Not that I presume to insinuate, however, that some people may not think you perfection already — Jane Austen

(+ into) to gain acceptance for (e.g. oneself) by crafty or stealthy means

He insinuates himself into the favour of the chief

to insert or squeeze in (oneself, a remark, etc) somewhere

Fagin walked straight up the stairs … and softly insinuating himself into the chamber, looked anxiously about — Dickens

I will insinuate a few remarks at this point — William James