The Sunday Times reports that Leighton is proposing that the government, which owns 100% of Royal Mail, lend it £2bn. Part of this would be spent on buying shares from the government at the end of each financial year in a three-year programme.
Some of the cash would also go towards reducing Royal Mail's £2.5bn pension deficit.
The newspaper says Leighton has been sounding out government officials about the plan but has had to wait until after the General Election to get their full attention.
Labour pledged in its election manifesto not to privatise the company, but Royal Mail officials say there is enough "wiggle room" to allow the plan to go through.
The post-election cabinet reshuffle means that Leighton will be negotiating with former postal workers' union chief Alan Johnson. Johnson was appointed on Friday to head the Department for Productivity, Energy and Industry, as the former Department of Trade and Industry is now known.
The current leadership of the Communication Workers Union is opposed to privatisation of Royal Mail, including any employee share scheme.
All of Royal Mail's 195,000 employees are already set to receive a one-off share of profits under a bonus scheme linked to the company's full-year results, out next week.
It is widely expected that individual bonuses of over £1,000 will be paid because of the company making operating profits of over £500m.
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