The Daily Telegraph claims TNT, which is part of the privatised former Dutch state postal service TPG and has become Royal Mail's main rival in the UK postal market, is the favourite ahead of Deutsche Post company DHL.
The newspaper reports that ministers launched the search for private investment in Royal Mail when it became clear that the independent Hooper report on the future of the universal postal service would recommend the involvement of the private sector.
Details of the report, which is expected to be published today, were leaked to the press at the weekend, revealing the depth of Royal Mail's financial problems. It noted that the Government was still opposed to its privatisation in line with the Labour Party manifesto commitment.
Selling a third of Royal Mail will allow the Government to claim that it is not embarking on full-scale privatisation.
There is likely to be considerable political opposition to the move from several quarters including the Communication Workers Union, which is threatening to strike in protest at any move towards privatisation, and backbench Labour MPs.
Opposition could also focus on indications that taxpayers will assume direct liability for Royal Mail's £7bn pension deficit to attract the private investment.
The Hooper report is expected to leave no doubt that radical action is needed to preserve the universal service.
It says the company lacks the money to invest in automation because it needs to make large payments to top up its pension scheme deficit of £7bn, which has doubled over three years.
Operating profits have also fallen over the past two years due to changes in its business environment.
While competition with private postal operators has sapped £100m in that period, this is dwarfed by the £500m impact of customers using digital instead of postal communications.