Sullivan also admitted any January transfer business was dependent on players being sold which will mean a quiet window for Slaven Bilic's side if it is reliant on the sale of the club's record signing.
If the Hammers do want to do business they will have to drastically lower their valuation to a more realistic figure like £8 million for any other clubs to even consider a bid.
After all that is exactly what they have had to do with winger Matt Jarvis, who was signed for over £10 million to provide the ammunition for the big Geordie.
Jarvis struggled throughout his time with the Hammers and was loaned out to Norwich City this season who have since completed a permanent 2.5m deal which will be rubber stamped when the transfer window opens.
That's an £8m loss so the club are clearly keen to raise some funds from somewhere.
An expensive plan B for a club like West Ham
When Bilic has a fully fit squad - a rare luxury of late - Carroll is very much the plan B behind star striker Diafra Sakho and is less suited the manager's free-flowing attacking style than the Senegal star.
Recent performances since his latest return from a long-term injury have been criticised by fans who expect more.
The striker showed he can make an impact from the bench once again, though, when he helped the Hammers come from 1-0 down to beat Southampton 2-1 on Monday night.
But there are a number of key reasons why Carroll won't be going to Newcastle, or anywhere else this January, unless the Hammers lower the asking price.
Who would risk £34 million deal?
He has spent more time in the treatment room than on the pitch since then, much to the frustration of everyone at the club, including the men who sanctioned the deal, Sullivan and his fellow co-owner David Gold.
According to the Telegraph's report, Carroll's wages are an eye-watering £90,000 per week, meaning the remaining four years of his contract are worth... wait for it... over £18 million in salary alone.
So West Ham reportedly want to recoup the £18m fee with the buying club knowing they would also have to commit a similar amount to wages making the overall deal a staggering £34 million.
So Carroll will not be going anywhere even if Newcastle show an interest as the St James' Park outfit would have to splash out almost the same amount in total they sold him for to Liverpool in the first place back in 2011.
Worth the risk for Newcastle?
And why would they take the risk?
No-one doubts Carroll's all round quality and work ethic - after all it is why fans and clubs have been so patient with him over the years.
The words 'lengthy injury' have time and again cast a shadow over the 26-year-old's career and restricted him to just 60 games and 16 goals in over three years.
His latest injury, back in January, kept him out for eight months, but it is nothing new. Carroll has been troubled by injury throughout his career, playing just 44 times for Liverpool over two years after signing for £35 million.
Even at Newcastle, he struggled to play on a regular basis, racking up just 80 appearances in the five years after making his debut for the Magpies.
Then when West Ham have needed him most in recent weeks he inevitably got injured again, missing draws against Swansea City and Aston Villa.
Would a club really be willing to risk such a massive outlay on a player with such a poor injury record, particularly one like Newcastle who find themselves in such a precarious state?
Perhaps his game changing performance against the Saints will convince clubs he is worth that risk.