Hull chief executive James Rule has joined chairperson and part-owner Kath Hetherington in Australia as they attempt to lure Puletua, Dragons centre Matt Cooper and Canberra halfback Michael Dobson to England.
They will be in the country for two weeks talking to NRL players and raising the club's profile.
But Puletua's manager Allan Gainey said on Tuesday his client was bound for St Helens if Hull "didn't get their finger out".
Gainey said he hoped to announce a Saints deal as soon as this week.
"I haven't met with anyone from Hull and we are very close to completing a respectable deal with St Helens," Gainey said.
"They (Hull) are going to have to pull their finger out very quickly if they are serious.
"If I don't hear anything from them in the next 24 hours by the end of the week I will have this all tied up."
Puletua is off contract at the end of 2009 but the Panthers have indicated the former Kiwi international will be released if he wants.
The Raiders have hinted that they will let the off-contract Dobson leave for England if Hull comes calling.
Hull coach Peter Sharp has targeted Dobson as a replacement for his injured playmaker Richard Horne.
"He's looking around over there so he's got to come back and talk to us and let us know," Canberra chief executive Don Furner said of Dobson.
"As far as I know he has received interest but no offers.
"If he can get himself a good deal over there we might look at letting him go but I have to talk to the coach about that."
Hull CEO Rule said the club had other players in their sights if Dobson was not available.
Rule said he would meet with Dobson's manager David Riolo "in the next few days" and representatives for Puletua and Cooper.
The Bulldogs have also sent feelers out for Cooper but incoming Dragons coach Wayne Bennett is believed to have told club officials keeping the centre is a priority.
"We want to put a marker down, raise the profile of Hull FC and sell the vision we have at the club - the best way we can do that is face to face," Rule told The Hull Daily Mail.
"I'll be speaking to Michael Dobson's agent, but I will be speaking to many agents.
"This trip was organised well in advance and we haven't flown here to sign one player, we have come for the long term benefit of Hull FC."