I have been in various positions in the LDS Church where I was responsible for "issuing a calling" - but I always have preferred the concept of "extending a calling". The difference can be subtle, but I believe it is important.
"Issuing" implies something being given as a matter of course that cannot be refused, while "extending" implies offering something that can be "retracted" if the person to whom the offer is being made doesn't accept it. Since I don't believe callings must be accepted (either reflexively or even after contemplation), but rather that everyone has the right and responsibility to ascertain whether or not they should accept any particular calling, it is important to me to choose words that allow the person to not feel obligated to accept. At the most fundamental level, I am wary of any phrasing that, even if only in theory or subtly, abrigates personal agency and pressures others to accept decisions I (and/or a presidency in which I am serving) have made regarding them.
Also, I never claim inspiration when I issue a calling - except in cases where I believe I have been inspired in very obvious and unmistakable ways, and those cases have occurred. In some cases, the genesis of the calling has been powerful revelation. Generally, however, I simply say something like, "I would like to talk with you about ______," I usually do this even in those cases where I believe the decision to extend the calling truly was inspirational or revelatory - although I have deviated from this general rule occasionally, when I have felt prompted to do so.
Likewise, I have heard far more leaders say, "We would like to extend this calling to you," than, "The Lord would like you to serve in this calling." I've heard the latter only rarely. I know there are leaders who phrase the extension of callings that way as a general practice, and I think they generally believe it, but I believe my own approach happens more often than claiming all callings are the result of undeniable revelation.