The former merger partners both have a 15.2 per cent share of the UK grocery market according to the latest Kantar figures, behind Tesco on 27.3 per cent and ahead of Morrisons, which has a 10.4 per cent market share.
Chris Hayward, consumer specialist at Kantar, said it is difficult to say whether Asda or Sainsbury's will pull ahead over the coming months.
"It feels like it will be close. Both are in that difficult middle ground where growth is hard to come by," he said.
"I think it is whoever is able to take best advantage if we have a long, hot summer and the events that are coming up. Whilst we haven't got the World Cup, we have got the Cricket World Cup, we've got Wimbledon, we've got the Netball World Cup in July."
Mr Hayward said that shoppers are weighed down with uncertainty and this will affect shopping habits.
"Brexit is not going to make me change my habits, but if I am uncertain - when the newspapers are telling me about things that are uncertain - I might feel I have less money in my pocket," he said.
"When I'm seeing British Steel and Jamie Oliver either going bust or running out of money, I am uncertain as a shopper.
"That means two things then happen. Shoppers either start going to cheaper supermarkets or they go to the same supermarkets and they buy cheaper products."
Sainsbury’s market share fell 0.5 percentage points to 15.2 per cent, with sales down 1.7 per cent, as it nears its 150th birthday.
Mr Hayward said: “Sainsbury’s is approaching its own anniversary a bit differently from rival Tesco – as well as reflecting on the past 150 years and including the Queen in celebrations, it is also looking forward and championing smaller brands with its Future Brands initiative. Sainsbury’s will be hoping that this strategy, combined with an increased level of promotions in recent months, will help win shoppers back.”
With the CMA ruling against the proposed merger now behind it, overall sales at Asda fell 0.2 per cent despite online growth of 10.7 per cent.
"It's a general trend where big four supermarket growth is more difficult whereas online is from a smaller base and you've just got more people doing it," said Mr Hayward.
"Asda are stronger around the family demographic. Those people have less time to shop."
Meanwhile, Morrisons remains Britain’s fourth largest supermarket with market share of 10.4 per cent down from 10.5 per cent this time last year.
Kantar said sales of beer, lager, ice cream and sun care fell substantially over the past four weeks as cooler weather hit discretionary spending.
The grocery market grew by 1.3 per cent during the 12 weeks to May 19, as the memory of a record-breaking May 2018 looms large over retailers.
Mr Hayward said: “This time last year we experienced the hottest May since records began and enjoyed major events like the royal wedding and FA Cup final – so we would expect this period to be challenging for all grocers when comparing year-on-year performance."
Sales of beer and lager, ice cream and sun care fell by 7 per cent, 12 per cent and 16 per cent respectively during the past four weeks as cooler weather hit consumer spending.
Mr Hayward said: “Growth of 1.3 per cent may appear modest when compared with last year’s 2.7 per cent. However, the sector continues to demonstrate resilience and volume sales remain unchanged from last year.
"We have two European football finals with British interest, the Cricket World Cup and the FIFA Women’s World Cup to look forward to in the coming weeks and retailers will be aiming to capitalise on these events and attract more shoppers throughout summer.”
Discounters Aldi and Lidl are worth a collective £344m more than this time last year and reached a combined record market share of 13.8 per cent.
Mr Hayward said: “The discounters continue to attract customers with nearly one million more households visiting Aldi compared with last year and an additional 630,000 shopping at Lidl. Around 75 per cent of growth at both supermarkets is coming from the ambient and chilled aisles as people opt for value in a greater proportion of their basket.
"The big four still account for about 68 per cent of sales. While the discounters are doing really well, let's not forget that seven of out £10 is spent in one of the big four."
Tesco's sales were supported by the performance of its own label ranges.
Mr Hayward said: “While Tesco’s overall sales were flat, its performance was the strongest out of the big four during the past 12 weeks.
"Exclusively at Tesco products continue to be popular and went home in a quarter of customers’ baskets while discounts linked to the retailer’s ‘100 years of value’ campaign delivered a further boost.
"Looking ahead, the supermarket will be hoping the decision to make lines from its Jack’s discount arm available in store throughout May will pique customers’ interest and generate incremental sales.”
Analysts Nielsen also reported grocery spending falling by 1.6 per cent in May in the absence of another spring heatwave.
Its figures show falling sales at three out of the top four supermarkets, and also at M&S, over the last 12 weeks, with Aldi and Lidl continuing to gain market share with double digit growth.
Mike Watkins, Nielsen's UK head of retailer and business insight, said: "Consumer confidence remains unchanged and with the cost of household bills continuing to rise, and 40 per cent of households feeling insecure about their finances, the economy remains the number one concern for shoppers.
"The result is that shoppers remain cautious about their grocery spend as they look to manage their overall household budgets."