Abbott, 49, who replaced John Ward in November 2008, stated earlier in the week that talks were near to completion and that there would be no announcement until all three had reached agreement.
Cumbrians' managing director John Nixon told the club's official website: "Discussions have now been concluded with the management team.
"All parties have agreed a one-year contract, in principle, and it's now just a question of completing the paperwork.
"Finalising that part of it takes a matter of hours, but we're in a situation where they are all away on courses or on family breaks.
"The manager and Tony are due back in Carlisle next week, so we'll be able to get that sorted, but Graham won't be back with us until he finishes his coaching course and his summer break at the end of June."
Abbott arrived at the club from Leeds' academy as first-team and reserve-team coach in July 2006 and was installed as caretaker-boss when Neil McDonald left the club at the start of the 2007/08 season before succeeding Ward a year later.
Kavanagh, who initially joined the club as a player on loan from Sunderland in 2008 before making the move permanent, was player/coach before stepping up to the number two's role.
Former goalkeeper Caig started his career at Brunton Park and went on to play for Blackpool, Charlton, Newcastle and Hibernian before returning to the club, initially as goalkeeper coach, during the 2009/10 season.