Buying CWW, which has issued three profit warnings since it split from the former Cable & Wireless in March 2010, would give Vodafone fixed lines that could relieve pressure on its wireless network and strengthen its position in the corporate telecoms sector.
But turning around the fortunes of CWW, which has suffered from a faster-than-expected drop in voice revenue, intense competition in data traffic and cuts in government contracts, is a challenge that has cost two chief executives their jobs.
Macquarie analyst Guy Peddy said he saw the extension of talks as an indication that discussions were advanced, and put a likely price on the deal of 40-50 pence a share.
CWW confirms advanced discussions with Vodafone are ongoing with a view to establishing whether or not they might result in a formal offer, CWW said in a statement issued minutes before a 1600 GMT deadline imposed by the takeover panel.
A banker familiar with the telecoms sector, but not involved in the transaction, said the extension had likely come about because Vodafone had offered less than 40 pence and CWW was keen to negotiate further to prevent a deeper fall in its share price.
CWW shares, which fell as much as 17 percent after Tata walked away, recouped some of their losses after sources told Reuters that Vodafone would either bid or request an extension.
They closed down 8.2 percent at 34.05 pence, valuing the group at about 936 million pounds ($1.5 billion).
Vodafone declined to comment on the discussions. It now has until 1100 GMT on Monday, 23 April to submit a bid or walk away.
The announcement Vodafone had been granted more time came just after the London market closed.
Marcus Allchurch, telecoms M&A specialist at BDO LLP, said Vodafone, in light of the recent withdrawal by Tata, would have insisted as much as possible on taking the weekend to hone its thoughts around an offer.
It's still not clear whether Vodafone will even make an offer, and if they do, much will depend on their assessment of the achievable synergies, he said.
In the absence of any competition for CWW, the trick will be paying enough to secure the asset and no more.
It has not been confirmed how much Vodafone Chief Executive Vittorio Colao would be willing to bid, but the company has said it would pay for a deal with cash.
The company is flush after selling minority assets in China and France in recent years, but Colao has set a disciplined approach to acquisitions.
Losses racked up over the last few years by CWW could be used by Vodafone to reduce the tax bill on its profits earned in the UK, analysts have said.
According to Bernstein Research, the C&W losses amount to 5.2 billion pounds, resulting in a potential 1.2-billion pound tax saving over a number of years.
($1 = 0.6226 British pounds)
(Additional reporting by Kate Holton, writing by Georgina Prodhan; Editing by David Hulmes)