Barrow joint-boss Dave Bayliss told BBC Radio Cumbria: "We are hoping to sign him up.
"He's a big lad and he's looked after himself. He's as fit and as quick as anyone at the club."
London-born Austin has made nearly 400 first-team appearances in a career that began with Saffron Waldren Town in 1992.
The lad is awesome and would be a great acquisition if he signs for us
Dave Bayliss
He has also had spells with Leyton Orient, Lincoln City, Barnsley, Brentford, Cambridge United, Kettering Town, Bristol Rovers, Swansea City and Chesterfield.
"Kev can play left-back and is an awesome centre-back as well," Bayliss said. "That could be a blessing in disguise as Paul Jones is struggling with his Achilles at the moment.
"We are looking elsewhere for a new left-back anyway and that's getting very close to getting the lad we want."
Bayliss is also confident that Austin's physical presence will prove a major benefit to his side if he can agree a deal for the 6ft 1in defender.
"He's a big lad, a beast," Bayliss continued. "You can't see a single roll of fat on him, he's looked after himself... the lad is awesome and would be a great acquisition if he signs for us."
Source: BBC Sport
Source: BBC Sport