The two groups had summertime talks aimed at creating a third superstores force but dropped the idea after soundings in Westminster indicated the Government would oppose such a deal.
But the stock market believes Asda has since put together a powerful case which it thinks could convince the Government that the get-together, perhaps with certain modifications, should be allowed.
Asda is thought to be keen to point out that a third force could be a more powerful influence in forcing down prices than the two chains operating independently.
A deal would create in some respects the largest supermarket chain in the country, removing Tesco from top spot. Asda's interim figures are due this month and some believe they could be accompanied by a Safeway announcement.
It has made it clear it is seeking acquisitions and has shown it is prepared to move away from its traditional operations by attempting to buy a motorway services chain. Safeway is obviously its preferred option.
Turnover in the two groups was heavy. Safeway moved ahead 1.75p to 323.5p. Last month the shares suffered a sharp reverse when it was revealed profits might not be much different from last year's pounds 420.6m.
Before the warning, Safeway was riding at around 400p. The shares have since been down to 319p. Their weakness could persuade Asda the chain is ripe for attention.
The Government's controversial savings changes, with supermarkets likely to be at the forefront in selling the new Individual Saving Accounts, also helped the sector, with J Sainsbury 4p firmer at 501p and Tesco up 13p to 495p. Banks are also in line for ISA sales success; Halifax rose 19p to 714.5p.
The arrival of ISAs will not be particularly rewarding for many diligent savers, who will face unexpected tax liabilities, and the more traditional saving groups. Unit trust groups Perpetual and M&G were hit, falling 177.5p to 2,420p and 42.5p to 1,352.5p respectively.
Footsie ended near its day's best, up 55.8 points to 4,977.6. The possibility that the Monetary Policy Committee may be tempted to lift interest rates again caused some anxiety but the overnight performances of New York and Hong Kong offered reassurance.
National Westminster Bank duly announced its departure from investment banking with the bulk of its equities business going to Bankers Trust and Deutsche Bank. The shares edged ahead 13p to 898p.
Hambros, the merchant bank, was 16p higher at 267p on hopes of a Societe Generale strike.
BICC, the cables and construction group, jumped 10p (after 15.5p) to 162p after Henderson Crosthwaite held an investment dinner at London's Howard Hotel. The company put on a positive performance and the share reaction recharged some of the old takeover stories with talk of Continental strikes.
Henderson was also at the Savoy Hotel, hosting a dinner for computer group Alphameric, up 2p to 53.5p.
Engineer Siebe rose 38p to 1,163p on its interim figures and today's analyst visit to its US operations. An upbeat trading statement lifted Williams 9.25p to 329.25p.
Nycomed Amersham strengthened 32.5p to 2,057.5p after the US Food & Drug Administration cleared its Teslascan treatment for liver disease.
Merrydown, the drinks group, added 15p to 65.5p on the prospect of bid action and Eldridge Pope, the pubs chain, put on 24p to 294p on its new votes-for-all policy. Continuing bid hopes lifted Zetters 6.5p to 131.5p.
Ramco Energy added 17.5p to 795p. There is talk it is on the verge of clinching an intriguing deal in Azerbaijan where it already has extensive interests. Earlier this year it failed in a bid to take over JKX Oil & Gas when Ukrainian interests moved in and blocked what had appeared to be a successful bid.