The 26-year-old QPR striker has been linked with a host of Barclays Premier League clubs over the summer after scoring 18 goals as the Hoops dropped out of the top flight last season.
West Ham are one of several sides who have reportedly looked to make a move for Austin, who was called up by England manager Roy Hodgson for the final two internationals of the season, but QPR have not budged from their £15million valuation.
But, when asked on the Knees Up Mother Brown podcast if Austin was a target, Sullivan was bullish about the former Burnley striker's injury record.
"They say he has no ligaments in his knee, who knows?," Sullivan said.
"To sign a £15m player is a big risk
He could go on for years, but knowing our luck his knee will go in his first game and that's the end of it.
"If we had £100m to spend we may say 'we'll spend £15m and gamble one-sixth of our budget'
But it's not one-sixth of our budget, it'd smash our budget to bits.
"And he didn't keep Queens Park Rangers up
If he'd kept them up single-handedly you might say it was different - and a few of those goals were penalties."
Austin released a personal statement on Twitter in reply to the Sullivan comments, defending his fitness and his professionalism.
"I feel I have no option but to address the inaccurate, misleading and uninformed innuendo about my physical condition that has been raised today by an individual who is not privy to my personal health history," his statement read.
"It is one of a number of inaccurate reports about my so-called injury problems which have been made over the summer.
"For the record, there is nothing wrong with my "ligaments", as has been suggested
My strength and performance in pre-season has been excellent and with two goals in my last two games I don't think there is any doubt that my match sharpness is as good as ever.
"I scored 18 goals in the Premier League last season, which would not be possible were I feeling discomfort or pain.
"Like many professional footballers, I have the legacy of injuries picked up over my career but the effect on my day-to-day training and on matchday is non-existent.
"For a senior figure at a Premier League club to insinuate that I could break down at any moment is an outrageous slur on my professionalism and the work that has gone into making me the footballer than I am today
I am fit, strong and looking forward to Saturday's home game against Rotherham."
The Hammers are keen to add a forward to their squad with Andy Carroll, Enner Valencia and Mauro Zarate all nursing injuries - with boss Slaven Bilic admitting it is a striker he is most likely to bring in before the close of the window.
It is understood West Ham had a list of up to 12 forwards who they were looking to bring in, but it appears Austin is not one of them as QPR look to hang on to their main asset in their battle to secure an immediate return to the Premier League.
Source : PA
Source: PA