The QPR striker has been linked with Newcastle United and West Ham United.
Last week, West Ham United co-chairman David Sullivan told Knees Up Mother Brown podcast that the club did not make an approach for Austin due to a previous concern with his knee.
“He (Austin) failed a medical at Hull a couple of years ago," Sullivan said in the KUMB interview. "They say he has no ligaments in his knee, who knows? To sign a player for £15m is a big risk.”
Sullivan subsequently clarified his comments about the 26-year-old, telling KUMB: “I fully accept that day to day he is fine and that his recent form is fantastic. The question is whether past injuries will impact on the longevity of his career.”
While not directly referring to Sullivan, former Spurs and Liverpool star Redknapp has hit out at those claiming that Austin has a problem with a knee.
“People say he has a bad knee but, speaking from experience, he is fine,” the former Tottenham midfielder wrote in The Daily Mail.
“I couldn't train until Thursdays when I had knee problems but I watched Charlie training every Monday last year. I'd be amazed if someone doesn't try to sign him before the deadline."
Austin was one of the best players in the Premier League last season and arguably the best for QPR.
The Englishman did all he could to prevent Rangers from getting relegated to the Championship at the end of the 2014-15 campaign.
The former Burnley star scored 18 goals and created 23 chances in 35 Premier League appearances.
Austin had a pass accuracy of 66% and a shot accuracy of 67%, and won 86 headers, 15 dribbles and 16 tackles.
The former Swindon Town man has started the 2015-16 season on a positive note, scoring four goals and creating seven chances in four Championship appearances.
Austin has a pass accuracy of 70% and a shot accuracy of 46%, and has won 12 headers, three dribbles and five tackles.
The QPR forward is an in-demand player in the summer transfer market, with Newcastle United reported to be interested in his services by The Daily Mail, while The Daily Star recently stated that Tottenham want him.