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.
Carroll has been much improved of late, helping change the game in West Ham's favour in the comeback win over Southampton and scoring in the comfortable 2-0 victory over his former club Liverpool.
And the club may be tempted to try and push on in this most open of Premier League seasons by cashing in on relegation-threatened Newcastle's desperation for firepower so they can bring in some new quality players.
Selling Carroll would not only generate cash but also free up around £90,000 per week on the wage bill.
And that could enable them to launch moves for these three players.
The Daily Mirror report that the Hammers are pursuing a deal for Besitkas star Gokhan Tore, who may be familiar to some Premier League fans having previously been on the books at Chelsea.
One man who is certainly familiar with Tore is Bilic, who managed the Turkish international during his time at Besiktas, and it now appears that the West Ham boss wants a reunion.
The report states that the Hammers will pursue a January deal for 23-year-old Tore - a left footed winger who can play on either side - in a bid to lure him back to London. But he could set the Hammers back anything between £8 - £10 million.
Leeds United boss Steve Evans has vowed that he will continue to select Sam Byram in the side until the moment that the youngster potentially decides to leave Elland Road, according to the Yorkshire Evening Post.
The 22-year-old's current deal with the Whites expires at the end of the season, and with clubs like West Ham and Liverpool taking a recent interest in his progress it appears that Leeds may be forced into a tough decision during the January transfer window.
That decision - over whether to cash-in on Byram this month rather than let him go for nothing, bar the compensation that Leeds would be due, in the summer - has been made much more difficult in recent weeks.
The academy graduate would not cost the Hammers the Earth but there would need to be some movement out as Sullivan has admitted the club has reached its ceiling for wages under the Financial Fair Play regulations.
With doubts over the form and long term future of Arsenal loan defender Carl Jenkinson, Byram would add much needed competition to Slaven Bilic's defence.
The least likely of the three but Valencia striker Alvaro Negredo has been on West Ham's radar for a number of years.
The former Manchester City striker had established himself as one of the most feared number nines in the Premier League during his spell at City.
But injuries and a loss of form saw him shipped back to Spain in a £20 million deal.
And his club Valencia are reportedly keen to offload him so they can pay City what they still owe for the Spaniard, as reported by the Daily Mail.
City are due £10million this year as part of the deal and it is unlikely new Valencia boss Gary Neville will have any power to stop him leaving should the Hammers renew their interest as finances take over.
He would certainly make for a more mobile replacement should Carroll leave, but the question is would he be able to recapture his best form.