The talented 22-year-old was released with immediate effect by Manly Sea Eagles earlier this week sparking a battle for his services.
Huddersfield thought they were leading the way for the ex-Sydney Roosters star but Woolford said: “I’m not sure.
“One day we’re confident, the next day we’re not so sure.
“We’ll see how it plays out in the next few days but obviously we’re not the only ones interested in him.”
Giants do, though, remain “confident” Manly’s Akulia Uate, the Fiji winger they have been in talks with, will sign for 2019 and insist an ankle injury he picked up at the weekend will not scupper any deal.
Huddersfield seek to continue their recent positive form when they head to Hull KR on Friday.
They have won their last five Super League games - three since Woolford arrived from Australia - to reach the crucial eighth spot.
“It’s all going well,” said the former Newcastle Knights assistant.
“The town’s been good, the weather’s been good - people keep telling me it’s not always this hot! - and we’re winning games.
“Obviously, the way we started the season means we can’t really afford too many slip-ups; every game is really important and this week is no different.
“We have to try carry on as we have as we’ve built some momentum.”
The only change to Huddersfield’s 19-man squad sees teenager Sam Hewitt replace Dale Ferguson.
Woolford added: “The squad’s looking really good; we’ve a few boys in rehab coming back from injury but generally speaking, especially considering the injuries we had earlier in the year, things are looking really positive.
“We should have Ollie Roberts back this week and next week we might have Ukuma Ta’ai so that bodes well.”
Hull KR are in 11th and in desperate need of a run of wins themselves if they are to have any chance of avoiding the Qualifiers although, even then, their hopes are slim.
Still, they have ex-Leeds Rhinos and England half-back Danny McGuire returning in their 19-man squad following a blood clot.
He will play if he comes through tomorrow’s final training session unscathed while boss Tim Sheens is further bolstered by prop Robbie Mulhern and hooker Tommy Lee being fit again after their own injuries.
The trio replace Matty Marsh, Elliot Wallis and Aaron Smith in the 19-man squad.