The Raiders need to strike an agreement with Gagai before round 13 for him to be eligible to play for them this year, and any deal would include the 2013 season.
Hamilton coached Mackay-born Gagai in 2010 as part of the Junior Kangaroos' two-match tour of New Zealand.
He phoned Gagai late last week to enquire about his welfare, and informed him there may be possible openings in the Raiders squad.
The Knights appear to have the inside running thanks to Wayne Bennett, who helped nurture Gagai in Brisbane, while joining the Cowboys would allow him to return home.
''Absolutely [we're interested in Gagai], but it's early days yet and we haven't put an offer in,'' Furner said. ''My understanding is Newcastle are very, very keen, and if I was a betting person I'd say he'd go with Wayne [Bennett] there, but we're trying.
''A one-team town, and a smaller town, would suit him better than a city from what I understand.
''I haven't had any discussions with his manager [Col Davis], I've left it to Hammo [Hamilton].''
Gagai is currently playing Queensland Cup for Redcliffe, and Hamilton insists there are no question marks about Gagai's character.
''I'm more than happy with the person, he's a great kid, and he has leadership qualities about him,'' Hamilton said.
''He was very professional in everything he did [with Junior Kangaroos], he was the one pulling some of the other kids back into line.
''I do recruitment and look everywhere to be honest, and there's not a hell of a lot of experienced outside backs out there.
''What I've said to him is there's definitely an opportunity here for him, and this year.''
Hamilton believes Gagai's dramas in Brisbane may have stemmed from his failure to crack a permanent first grade spot.
Despite scoring four tries in seven NRL games last year, Gagai has been on the outer at the Broncos behind Josh Hoffman, Jharal Yow Yeh, Dale Copley and Gerard Beale.
''I think it may be down to him being a little frustrated he's not getting a go in first grade,'' Hamilton said.
''I know [head coach] Dave Furner has come under a bit of pressure, but he's helped produce a lot of these young boys and that could be an attraction for Dane.''