Offer On The Table Could Be Snubbed
Gillespie was hugely influential for the Blades in the first few months of last season but then received a four-game ban after being sent off against Reading in January and days later he handed in a transfer request.
The former Manchester United and Newcastle player did not return to the side until mid-March and was powerless to prevent the south Yorkshire club from dropping out of the Premiership, prompting the resignation of boss Neil Warnock.
Blades chairman Terry Robinson has made the offer even though the Blades don't have a manager.
"We would like to make an offer to Keith and, although the new man might have different ideas, at this stage we felt that we should try and keep him."
Gillespie came to Newcastle as part of the deal that took Andy Cole to Manchester United.
1995/99 K.R.Gillespie 121/25 14
Northern Ireland international Keith will be best remembered as the player who signing sent Andy Cole to Manchester United. "The deal wouldn't have gone through if I couldn't sign Gillespie" said Kevin Keegan. A lively influential winger, and his injury (ironically at Old Trafford) cost Newcastle the title in 1996. Moved to Blackburn Rovers. Northern Ireland international born in Larne. Took a punch off Alan Shearer after an argument in a bar when the club was on a short pre-season tour of Ireland. Supposedly Gilly upset a few drinkers with his antics and bad language, and Shearer took him outside and taught him some manners.