Paul Pester, chief executive of TSB, has previously said Lloyds will initially sell 30% to 50% of the business, through an initial public offering.
This is expected to include a retail offer for small shareholders.
Part-nationalised Lloyds is expected to launch the IPO on May 19 and complete it by the end of the month. This is despite recent rocky markets.
John Beard, who ran whisky company Whyte & Mackay until last summer when Johnnie Walker owner Diageo bought took over, is fronting a bid to buy his former employer.
The Sunday Times said that Mr Beard, who left in August after four years, has teamed up with private equity firm Lion Capital.
Competition chiefs have ordered Diageo to sell the Glasgow-based business because its Bell's brand overlaps with Whyte & Mackay's own label and branded blends.
Top private equity firms are thought to be in the running for a £4.1 billion package of quarries and cement plants being offloaded by Lafarge and Holcim to ease their planned merger past competition authorities.
The Sunday Times said that Holcim's Aggregate Industries arm in the UK, whose interests include cement operation at King George V Docks in Glasgow and Glensanda Quarry near Oban, is likely to be sold off.
Heineken is reported to be considering plans to sell up to 10% of the pub estate it acquired with its 2008 takeover of Edinburgh-headquartered Scottish & Newcastle.
The Dutch brewer has 1250 pubs in the UK and is thinking of selling around 100 of the poorest performing ones, the Sunday Times reported.