Company sources are said to have confirmed that meetings, first mentioned in the Sunday Times and subsequently followed up by Reuters, are currently taking place.
C&W is the UK's second largest fixed-line operator, and is expected to buy Energis for around £700m. Both companies have declined to confirm the reports.
C&W has £1.34bn in cash that could finance such a deal, which would net it some major blue chip clients including the BBC, IBM and Tesco.
The telco has itself been the subject of takeover talks when it was approached by Hong Kong carrier PCCW in February 2003.
"A merger [between C&W and Energis] would produce a firm generating about £2.3bn in annual sales, making it a substantial rival to BT," said Clem Chambers, chief executive at shares website ADVFN.
However, Dr Simon Rawling, global head of project management at consultancy PIPC, warned: "Such a deal would throw up some major integration and branding challenges. Many such integrations fail.
"The Morrisons/Safeway deal is a good example and the latest in a long line of mergers and acquisitions that have failed to maintain shareholder value."
The talks represent something of a rise from the ashes for C&W. Shares fell to a 10-year low in 2002 amid concerns over accounting practices after the company admitted that it had booked non-cash transactions as revenue.
After cutting thousands of jobs in the past few years C&W returned to net profit last year.
Energis has been a sale target since its debtor banks, to which it owed over £700m, took over the business when it became insolvent three years ago.
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