Sales at Sainsbury's were up 0.2% over the 12 weeks to April 22, while Asda's were up 1.4% compared with the same period last year, according to Kantar Worldpanel's latest market share data.
Both Sainsbury's and Asda dropped market share compared with this time last year - down 0.3 percentage points and 0.1 percentage points respectively.
Meanwhile, Tesco enjoyed a 2.1% increase in sales and Morrisons saw 2.2% growth.
Kantar said almost two-thirds of Asda's sales were outside London and the South East of England, in contrast to Sainsbury's 59% within the two areas.
Some 15.8 million households bought their groceries at Asda over the past 12 weeks, 500,000 more than shopped at Sainsbury's.
Fraser McKevitt, head of retail and consumer insight at Kantar Worldpanel, said: "This is a pivotal moment for the British grocery market. A merger between Sainsbury's and Asda would transform the traditional landscape placing nearly a third of market share in the hands of the joint supermarket giant, though the march of the discounters - and any enforced store closures - could impact this figure."
Overall, the British grocery market grew at its slowest rate since March 2017 at 2% - the result of lower grocery price rises. The like-for-like inflation rate is now 2.1% and is expected to fall further in the coming months.
Lidl became the UK's fastest-growing bricks and mortar supermarket with sales up 9.1%, while Aldi continued to experience strong growth with sales up 7.7%.
Nielsen retail figures show the unpredictable weather again disrupted grocery spending, with sales at the UK's leading supermarkets growing by 1.2% year on year over the four weeks to April 21 - the lowest rise since January last year.
In addition to the overall slowdown in the amount shoppers spent, the number of items purchased year on year fell 1.1% - the poorest growth figure for more than seven months.
Mike Watkins, Nielsen's UK head of retailer insight, said shoppers were still cautious about spending despite food price inflation slowing down.
However, Nielsen found that Asda's sales momentum continued, having the most improved year-on-year performance of the top four supermarkets with sales up 2.3%.
According to Nielsen, Asda's share of the grocery market stands at 13.9% and would mean a combined market share with Sainsbury's of 28.8% if the proposed merger went ahead - above Tesco's 27.1%.
Mr Watkins said: "Both retailers have over half of all households as shoppers every 12 weeks, while over three-quarters also visit Tesco. This illustrates how competitive the retail landscape is, which is why we're seeing further consolidation within the industry."
Sainsbury's sales grew 0.2% year on year, while Tesco's grew 2.1%, Nielsen's data shows.