The Dividend Access Share was acquired by the Government during RBS's rescue in 2008 which left it 81% state-owned by the taxpayer.
The share is valued at £1.5 billion on the Government books, although there have been suggestions that its worth has fallen.
Any purchase of the share would have to be approved by RBS's remaining non-Government shareholders so the bank is understood to be keen to get a rubber-stamp from investors this year.
Any payment for the share would only be made at the time that RBS wanted to start paying dividends, rather than when the deal was signed.
The purchase of the share is seen by bank executives and the City as one of several steps needed to be taken before it can be returned fully to the private sector.
The bank has already started paying dividends on its preference shares.
It has also exited the Government's Asset Protection Scheme designed to protect it against large loan losses.
In December, it cancelled the contingent capital facility with the Treasury, for which it had been paying £230 million a year. Two years ago, RBS chairman Sir Philip Hampton promised to restart dividend payments "as soon as possible".
RBS declined to comment.