The Labour peer and former Standard Chartered chairman, who served in Government under Gordon Brown, is said to be trying to bring together a consortium of investors and overseas funds to buy a £10 billion chunk of the bank, which is 39% state-owned.
Lord Davies is vice-chairman of private equity firm Corsair Capital, which it is reported would be part of the bid.
Institutional investors and sovereign wealth funds are also said to have been approached about joining the consortium, in a deal which would help the Government kick-start a full-scale privatisation of Lloyds, the Sunday Telegraph reported.
The Treasury and UK Financial Investments (UKFI), which manage the taxpayer stakes in Lloyds and the Royal Bank of Scotland, were thought to be aware of the plan, which Lord Davies had been working on for more than a year, according to the newspaper.
Last month, Chancellor George Osborne said in his annual Mansion House speech that the Treasury was "actively considering" steps to return Lloyds to the private sector.
The Government bought stakes in Lloyds and RBS at the height of the financial crisis in 2008. A sale in its stake of the latter, which is 81% owned by the taxpayer, was said by the Chancellor to be "some way off".
The Treasury and Lloyds declined to comment on the latest reports on Lloyds. Corsair could not be contacted for comment.