Town have revealed the 28-year-old has been given permission to travel on Friday for five days in order to undergo a medical and discuss personal terms - with the club reported to be Brunei DPMM FC, who are managed by former Blackburn boss Steve Kean and play in the Singapore S.League.
O'Donovan will then return to England on January 8 and make a decision over his future, which may still lie with the Cobblers in League Two.
The news comes as a surprise after O'Donovan signed a two-year contract in the summer, having arrived at Sixfields last January on an initial six-month deal following his release by Coventry.
The Irishman has made a total of 15 appearances this term, scoring twice, having struggled with injury, spending two months on the sidelines after a hernia operation.
Northampton caretaker manager Andy King told the club's official website: "I know his agent and he came out of the blue with this opportunity for Roy.
"I said, quite simply, there's a little bit of an injury thing at the moment. If it favoured the player then I'm for that, if someone wants to do something then they can do it, I'll move on to the next situation.
"He's going out to Dubai, it must be a contact his agent has got, and he's going out there to have a fitness test with a view to being offered a contract. And he'll either take it or he won't.
"I asked Roy if he wanted to make that choice, he said yes because he's also been in and out and his future is not certain, that's the way it's been here with Roy.
"Once he said he wanted to take the opportunity then I said fine, go and if it suits you then I'll speak to the chairman and see if that's something we want to take. If not, as soon as you get back then you're in the frame for selection."
O'Donovan has also previously played for Cork and Sunderland while he has enjoyed loan spells with Dundee United, Blackpool, Southend, Hartlepool and Hibernian.
Elsewhere, Northampton have confirmed Matt Heath has left the club following the expiry of his short-term contract.
The defender arrived in the summer but has made just seven appearances, the last of which coming in September.