Furneaux however has told the manager that he can have the one promised striker and no more.
The chairman told the Grimsby Telegraph that the club has a large hole in it's budget, with almost £500,000 of promised money not arriving as of yet.
All Division Two clubs were promised a £333,000 loan from the Football League following the collapse of the ITV Digital contract - this has not arrived.
A grant of £50,000 from the combined forces of the PFA and the Football Trust was promised - this has not arrived.
Following relegation from Division One, Town (along with Brighton and Sheffield Wednesday) were promised a parachute payment of £100,000 from the teams promoted to the Premier League. Whilst Portsmouth, Leicester and Wolves have seen their income increase up to the tune of £18million, nothing has arrived.
With this is mind, Furneaux is refusing to open the cheque book for Groves other than for a new striker.
Furneaux did however give hope that improved gate revenue and more share sales would allow the manager the cash he needs.
"The money that we allocated for players was clearly identified to the manager so that he could get the players he wanted," Furneaux told the newspaper.
"He knew full well that if we could sell more shares and get the crowds up there would be the potential for more.
"But the board are already financing losses that are anticipated for next year as well as money that has not been received.
"We need new money to come into the club if the manager is going to get his wish for new players, he knows that. If we sell more shares, he can spend the money.
"We gave him quite a good budget and if Laurens Ten Heuvel goes back he can have that money, otherwise he knows exactly where he stands.
"We haven't had the money for going down. We haven't had the loan from the Football League or the money for being in the second division, and there is no anticipated TV income. We have to be very careful."