Northern Ireland international Healy has turned down the offer of a new three-year contract at Deepdale and The Whites made an initial bid of a reported £100k for the 25-year-old.
Shaw said that offer was "laughable" and he has also been unimpressed by the improved offer, this time thought to be in the region of £200k.
Shaw told Preston's website: "This is totally unacceptable.
"Leeds came in with an improved offer but, again, it is nowhere near a figure we would even consider."
Preston are believed to want £400k for Healy but may have to settle for less due to the player being available on a free transfer next summer when his contract runs out.
Kevin Blackwell said: "Preston may have their valuation of David but business is business and if I do a deal it will be the best deal for Leeds United. The process is underway and hopefully we can come to some sort of agreement through common ground and negotiation.
"We have established a starting point. Preston happen to have a lad who is out of contract and we have offered what we feel is the market value at this level.
"There are guidelines as to how a player is valued in terms of what has gone before. Michael Brown went for £500,000 from Sheffield United last Christmas and he had scored 25 goals from midfield.
"Big clubs in England and Scotland were looking at him but he still went for just £500,000 because his contract was nearly up.
"I am looking to sign David for the long term so if I have to wait, then I will wait."
Shaw said that offer was "laughable" and he has also been unimpressed by the improved offer, this time thought to be in the region of £200k.
Shaw told Preston's website: "This is totally unacceptable.
"Leeds came in with an improved offer but, again, it is nowhere near a figure we would even consider."
Preston are believed to want £400k for Healy but may have to settle for less due to the player being available on a free transfer next summer when his contract runs out.
Kevin Blackwell said: "Preston may have their valuation of David but business is business and if I do a deal it will be the best deal for Leeds United. The process is underway and hopefully we can come to some sort of agreement through common ground and negotiation.
"We have established a starting point. Preston happen to have a lad who is out of contract and we have offered what we feel is the market value at this level.
"There are guidelines as to how a player is valued in terms of what has gone before. Michael Brown went for £500,000 from Sheffield United last Christmas and he had scored 25 goals from midfield.
"Big clubs in England and Scotland were looking at him but he still went for just £500,000 because his contract was nearly up.
"I am looking to sign David for the long term so if I have to wait, then I will wait."