Sky Sports understands a host of Football League clubs are tracking non-league hot-shot Scott Fenwick.
Middlesbrough boss Aitor Karanka has revealed that the club have opened contract talks with captain Grant Leadbitter.
The Cumbrians, who slumped to a fourth league defeat of the season at the weekend, lie third from bottom in the table after registering just two points from their first six matches.
Abbott, 49, was the third-longest serving manager currently in the Football League having been appointed as John Ward's successor at Brunton Park in December 2008.
Assistant boss Graham Kavanagh and youth team coach David Irons have been placed in temporary charge.
A club statement read: "The board would like to go on record with their acknowledgement and appreciation of the hard work Greg Abbott has done for Carlisle United since he joined the club as first team and reserve team coach in July 2006.
"He has maintained a very professional and close working relationship with everyone at the club and he has put his heart and soul into his stated aim of bringing improvement to every level and area of the business.
"The trips to Wembley and the year-on-year improvement in league position, prior to last season, were testament to his efforts and dedication to the job at hand.
"This has been a very difficult period and he has worked hard to find the players he felt would help to turn things round through what has been an unbelievable run of injuries.
"Unfortunately the continued dip in attendances, in direct relation to results here at Brunton Park, left all parties with no other option than to reach this conclusion.
"Greg Abbott leaves Brunton Park with our utmost respect and admiration and we wish him every success for his future in the industry."
Abbott, who made over 280 appearances as a defender for Bradford and more than 120 for Hull during a 14-year playing career, joined the Leeds coaching staff before moving to Carlisle as assistant manager to Neil MacDonald in 2006.
He was appointed caretaker-boss when MacDonald was sacked in 2007 and was then assistant to John Ward before taking on the role himself when the latter was sacked in December 2008.