Any bid would be almost certain to be made through Sun Life and Provincial Holdings, AXA's separately quoted UK offshoot, although AXA might have to step in to help fund the deal.
GRE, the smallest of the British composite insurers with a market capitalisation of around pounds 2.4bn, has embarked on an internal restructuring aimed at giving the group a chance of remaining independent in the medium term.
However, the City remains sceptical about GRE's long-term future and is still angling for a takeover bid for the group. Some big City investors believe GRE is too small to survive in a fast-consolidating insurance sector and are unconvinced about the prospects for growing the PPP healthcare business when the Government is committed to building up the National Health Service.
One of the main obstacles to a deal could be GRE's chief executive, John Robins, who is determined to keep the group independent.
AXA has made no secret of its wish to expand by acquisition in the UK. It is also keen to grow the UK shareholder base of Sun Life and Provincial, which is 72 per cent owned by AXA.
AXA has been seeking to build its position in home and motor insurance on the back of its sponsorship of football's FA Cup. GRE's pounds 300m UK and Irish motor book would give the firm much-needed bulk in a competitive market.
AXA chairman Claude Bebear admitted some months ago that he had been targeting a company in the UK quoted sector but had been put off by the high prices financial assets were fetching.
However, since then GRE's share price has fallen more than 30 per cent: the shares closed yesterday at 275p, barely in line with its net asset value.
Sun Life and Provincial has a market value of around pounds 4bn. AXA, its parent, is capitalised at pounds 25bn and would have no problem finding the cash to pay for the right acquisition
AXA's last major UK move was to merge its existing Equity & Law business with Sun Life, where it had acquired a controlling stake through the all- French insurance merger with UAP in 1996. AXA, which controls Equitable Life in the US and National Mutual in Australia, is also keen to build up its UK shareholder base. A merger with a UK player would be the obvious route to this goal.
Executives at AXA were last night unavailable to respond to questions about a bid for GRE. A spokeswoman said later: "This is market rumour. It is not our policy to comment." A senior executive of GRE said he was not aware of any approach from AXA.
Analysts said yesterday that the injection of Guardian's small life insurance business into PPP, the healthcare business acquired earlier this year, had made GRE more rather than less attractive to any predator.
"The restructuring is too little, too late. Sun Life will be buying into a business where someone has already done the hard work for them," said Matthew Wright, insurance analyst at Daiwa. "It is not an overly compelling deal for them, but there aren't any massively obvious fits left."Reuse content