The stopper spent half of last season at Varese in Serie B.
It is being reported by The West Australian that both Newcastle and Bournemouth are hopeful of bringing in the talented stopper.
Newcastle are currently without Tim Krul through injury so another stopper would not go amiss and Bournemouth's own Australian Adam Federici is injured and has also struggled with Premier League football, whilst Artur Boruc has been in his usual enigmatic form.
Birighittii is a 24-year-old stopper who has one cap for the Australian national team and spent part of this year on loan in Italy - where he has heritage.
He spent the second half of last season on loan with Varese in the Italian Serie B and is a very talented option for either side.
He has previously had a trial with Bayer Leverkusen and received international attention after receiving a sickening boot to the face in a game earlier this year.
Birighittii is only contracted with the Jets until next June so would be available for quite a reasonable fee in January or on a Bosman deal for next summer.