Amongst the list of players with unsure futures are top scorer Michael Boulding, vice captain Alan Pouton, Phil Jevons, Stuart Campbell and Mike Edwards.
Officialy any player with 6 months or less on his contract can open talks with other clubs from tomorrow (1st January).
Despite the threat of losing such important players, Furneaux insists talks won't open until the club knows what division it's in and how much money it can spend.
Furneaux also believes holding back contract talks with prove a good incentive for the players.
"There are players here who we would love to offer contracts to. But until we are sure the money is in place to make them an offer that they would be interested in, then we won't do so," Furneaux told the Grimsby Telegraph.
"Making an offer at the moment would only dis-incentivise them, and give other clubs the idea of how much they need to offer.
"If we tell players they might be on less money next year will we still get the effort we are currently getting until the end of the season?"
Pouton, Campbell, Jevons and Rowan are certain to take a pay cut if they are to stay, and Groves admits the club's future must come first.
"First and foremost you have to look at the financial situation at the club. Everybody wants to keep their best players and so do we," said the manager.
"From a financial point of view what this club has managed to do is stay out of administration. It's been supported heavily by the directors so the club's survival has to come first.
"That will have an effect. We have to look at our average gates and the revenue and that will all have a bearing on the contracts that can, and will, be offered to our best players.
"I think everybody wants to keep the best players but there is a lot of football to be played this season. We'll have to see how things develop before we start entering into negotiations.
"If players are allowed to get to the first of Janaury in the last year of their contracts then the laws state that they're allowed to speak to clubs.
"There's nothing you can do about that. But they're still under contract to us and they've still got a job to do for Grimsby Town Football Club.
"Ultimately if they want to improve they have to play well and hopefully their futures will be here.
"It works both ways. A successful side will highlight individuals that are in that team. A successful side would obviously help us to keep our better players."