Redknapp has so far failed in his quest to add to his defensive options, being turned down in approaches for Manchester City’s Joleon Lescott, West Brom’s Jonas Olsson and Everton’s Sylvain Distin.
Despite continued interest in Tottenham’s William Gallas, Redknapp is said to have made a move to sign Bramble from Sunderland.
And the Sunday People claims: "Sunderland would be keen to do a deal rather than let Bramble run down his contract to its conclusion in the summer when they are obliged to offer new terms if he starts more than five games this season."
But with Martin O'Neill struggling to put out two fit central defenders, due to injuries, the deal isn't as staight forward as the People seem to make out.
Granted, 31-year-old Bramble has only played a bit-part role for Sunderland this season following the arrival of Carlos Cueller in the summer.
But he did start in Saturday’s 3-0 win over West Ham – his third start of the season.
"Sunderland are set to bring in Senegal defender Kader Mangane on loan for the remainder of the season which could pave the way for Bramble’s exit from the Stadium of Light."
Martin O'Neill vows to treat FA Cup seriously on Sunderland's 40th anniversary.
O’Neill insists that he owes it to Wearside supporters to send out a powerful side in Tuesday’s third round replay against Bolton as the club try to emulate the deeds of Wembley heroes such as Jim Montgomery and Ian Porterfield.
The Black Cats boss chose not to ring the changes for the 2-2 draw at the Reebok and named his strongest side as Sunderland’s 2013 FA Cup quest began.
Martin O’Neill: “When I heard that we were taking 5,000 fans down to Bolton, I don’t think it would have gone down well if I had fielded a weakened side or included a couple of very young lads.
“The competition is so important to us and we want to try and stay in it.
“Also, while it’s not the main reason by any stretch of the imagination, it is 40 years since the club had that magnificent cup run and lifted the trophy, so we have to pay that some respect.
“The busy Christmas period has taken its toll even on those clubs that carry big squads.
“But even if I had had more players, I wouldn’t have done it because this is the FA Cup and it is an important competition.”
Bolton/Sunderland v Cheltenham/Everton
QPR/West Brom v Sheffield Wed/MK Dons
Huddersfield Town v Leicester City
Reading v Sheffield United
Southend/Brentford v Chelsea
West Ham/Manchester United v Fulham/Blackpool
Crystal Palace/Stoke v Manchester City
Brighton v Swansea/Arsenal
Leeds/Birmingham v Tottenham
Millwall v Aston Villa
Middlesbrough v Aldershot
Hull/Leyton Orient v Barnsley
Derby v Blackburn
Macclesfield v Wigan/Bournemouth
Oldham v Mansfield/Liverpool
Norwich v Luton