Kavanagh, 39, emerged as an early favourite to replace former Hartlepool manager John Hughes last week following the boss' sacking, although Middlesbrough under-18s coach Cooper, 46, is reported to have verbally accepted an offer.
However, Carlisle managing director John Nixon expects Cumbrians boss Greg Abbott, assistant Kavanagh and coach Tony Caig to sign their new contract offers shortly.
Nixon said no approach had been made by Hartlepool to speak with Kavanagh about the current vacancy at Victoria Park.
"All three members of staff have had their offers and the indication is that they all want to announce their agreement together," Nixon told Carlisle's official website.
"There are one or two items up for discussion, but I don't think there is anything too significant in there.
"It's all standard for this kind of thing and we're expecting to hear back very soon."
Former Middlesbrough, Stoke, Cardiff and Sunderland midfielder Kavanagh joined up with Abbott at Carlisle as player/coach in January 2009.
Nixon added: "Graham is under contract until the end of June and, as far as we are concerned, he is now in the process of agreeing to a one-year extension to that.
"With that in mind, and with the type of character he is and the excellent working relationship we have with Hartlepool, I've no doubt both parties would have contacted us had there been any substance to these rumours. We've had no contact at all about this issue."