Discussions between them have been going on for some time, although it is unclear when a deal might be concluded.
A City source said the waste business, which handles street-cleaning contracts for local authorities, could be worth in the region of £75m.
Firm news of the sale would represent a rare boost to Serco, which has been forced to issue a string of profit warnings and part company with Chris Hyman, Mr Soames's predecessor.
A £550m rights issue to shore up its balance sheet is planned for next year, and would prevent Serco breaching borrowing agreements with its banks.
Mr Soames has announced plans for the sale of a trio of non-core businesses, including the environmental services division, which is being overseen by Rothschild.
The company has appointed Citi to handle an auction of Intelenet, a business process outsourcing provider acquired by Serco in 2011 for £385m.
Alongside The Listening Company, which manages customer contact services for clients, it could fetch roughly £500m, according to analysts.
Livingstone Partners, another advisory firm, will supervise a sale of the company's leisure unit, which manages around 70 facilities across the UK.
The private equity giant Carlyle is interested in buying some of the assets which have been put up for sale.
The environmental and leisure businesses account for only around 3% of group revenues and are collectively worth far less than Intelenet.
In a shock profit warning last month which wiped 30% off Serco's share price, Mr Soames said the news was "a bitter pill", adding that his initial review of the business had involved "much turning over of stones".
He said that the company's future would be as a focused supplier of services to governments in areas such as justice and immigration, defence, transport, citizen services and healthcare.
The profit alert was the latest in a string of problems for Serco, which had been hit by a ban on winning new Government contracts after it emerged that some work. monitoring prisoners had been billed for fraudulently.
Serco eventually agreed to repay tens of millions of pounds to the Government after intense pressure from ministers.
On Wednesday, Serco announced that it had won a five-year contract from the Australian government to help run immigration detention centres, which could be worth up to £1bn.
The deal covers a range of services, including transport, accommodation, welfare and education in seven Australian states and territories, and helped push Serco shares up by as much as 6% in morning trading.
Serco and Rutland both declined to comment on the transaction involving the waste services business.
Sources: Sky News, CNS, PBROTRIB
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