Black Cats officials turned down Liverpool's offer of a short-term deal for the Trinidad & Tobago international, with manager Steve Bruce insisting that he would not allow any of his senior players to leave on loan this month.
However, Liverpool boss Rafael Benitez is ready to make a new £12m offer in an attempt to kick-start his side's faltering season and bolster a strike force that has been deprived of Fernando Torres' services for up to six weeks.
The Reds boss has been able to raise more than £6m this month through the sale of Andrea Dossena to Napoli and Andriy Voronin to Dinamo Moscow, and Liverpool's American owners have pledged to double that figure in order to enable Benitez to sign the striker Liverpool have arguably been lacking for the last two seasons.
Sunderland chairman Niall Quinn insisted that Jones would not be leaving the Stadium of Light when he appeared on Sky Television on Sunday, but those comments came in the wake of Liverpool's failed attempt to engineer a loan deal.
A formal £12m offer could lead to a change in attitude from the Black Cats, as it would generate funds for manager Steve Bruce to spend this month.
Bruce is looking to sign two defenders, a midfielder and a striker, and while there are funds available whether anyone departs this month or not, his options would be greatly increased by the receipt of an eight-figure fee.
However, Sunderland turned down an offer of £15m from Tottenham last January, and while Jones' form this season has been patchy, it remains to be seen whether £12m will be deemed a high enough fee for one of the most highly-rated players on the Black Cats' books.
Jones is not the only Sunderland player attracting interest, as Tottenham boss Harry Redknapp is ready to make a renewed attempt to sign Anton Ferdinand.
Redknapp is poised to table a straight cash offer of around £4m for the centre-half, but has also hinted that he is willing to consider a swap deal involving right-back Alan Hutton.
A similar deal was mooted in the closing stages of the August transfer window, only for both parties to pull the plug on transfer-deadline day.
Tottenham's defensive problems have worsened since then, though, and Redknapp is hoping to finally secure Ferdinand's services before the transfer window closes on February 1.
Interestingly, sources in the capital claim Ferdinand was at White Hart Lane on Saturday to watch Tottenham draw 0-0 with Hull City.
Redknapp is prepared to include Hutton as part of any deal for the former West Ham ace, and with Sunderland needing to bolster their defensive ranks, a deal that was first proposed five months ago could finally come to fruition.
Hamburg's Guy Demel remains an option, but Sunderland's pursuit of the Ivory Coast international is complicated by his presence at the African Cup of Nations in Angola.
It has been in the news that there is an interest of a club from England in the services of Guy Demel, said Hamburg chairman Bernd Hoffmann.
There has, however, not been any bid for him coming in. Looking at the current facts, Guy Demel is still part of our Bundesliga squad.
Meanwhile, Sunderland goalkeeper Craig Gordon will make his eagerly-awaited return from injury in Thursday's reserve game with Manchester United.