The research firm said wet weather has dampened sales and the supermarket sector is facing tough comparisons with last year when the heatwave and the run up to the men’s FIFA World Cup boosted sales.
Bradford-based Morrisons' sales fell 0.5 per cent over the quarter and Leeds-based Asda's sales were down 0.1 per cent. Tesco saw flat growth and Sainsbury's sales fell 0.6 per cent, halving the rate of decline it registered last month.
Fraser McKevitt, head of retail and consumer insight at Kantar, said sales were down or flat at the big four for two reasons.
"Firstly, look out of the window. It's a tall order for them to keep up with the heatwave we had last summer," he said.
"We had the men's Football World Cup last year, which boosted beer and crisps sales as well as barbecue products. There were a lot of deals in store for the World Cup last year.
"Secondly, the rate of grocery inflation has come down again and is down to 1 per cent now. That takes the overall market growth rate down, which is of course great news for consumers because we spend about £4,000 a year for the average household in the supermarket, but it does make it a bit hard for the main supermarkets, particularly those big four, to grow."
Kantar said Asda will be buoyed by the performance of its online arm, which enjoyed double-digit growth of 10 per cent.
"Asda is putting an effort into rolling out its online sales. Click and Collect is doing pretty well for them," said Mr McKevitt.
"Quite a lot of people who are put off by waiting in for a delivery find it convenient to do their shopping on the internet and then go and pick it up.
"If you just look at food, Asda saw modest growth. Ambient food sales were up 4 per cent. They are still cutting back on multi-buy deals as part of a move to the simpler pricing structure."
Asda’s share of the overall market fell by 0.2 percentage points to 14.9 per cent, while Morrisons lowered its share to 10.4 per cent of grocery sales.
"Morrisons have had a sticky few months," said Mr McKevitt.
"They are not attracting new shoppers through the door which is a little bit of a problem for them. They're not losing customers. The people they do have, their basket sizes are smaller. They need to find a way to reverse that and they need to find a way to attract new people.
"Obviously Morrisons have a number of irons in the fire that aren't necessarily captured in these numbers to do with their relationship with Amazon and selling their products elsewhere. So that could be an upside for them.
"One bright spot, which may be linked to elsewhere in the market where a lot of counters have been closed at their rivals, is Morrisons' fresh fish sales are up by 8 per cent in the 12 weeks."
Analyst Bruno Monteyne at Bernstein said Morrisons' performance was its worst since December 2016 and the latest figures compound the weak period last month.
"Morrisons performed about in line with Sainsbury's, which is a marked difference to its performance before the turn of the year when it tussled to be the best performer amongst the big four," he said.
"Food & Drink growth of minus 1.2 per cent was the worst out of the big four for the second consecutive period, whilst general merchandise growth was the best out of the big four at plus 4.3 per cent."
Kantar said overall grocery sales rose 1.4 per cent year on year during the 12 weeks to June 16. This marks a new milestone for the sector, which is celebrating a three-year trajectory of continuous growth dating back to July 2016.
However, the wet summer has hit sales.
"These challenges are reflected in typical summer categories," said Mr McKevitt.
"In the past four weeks sales of ice cream were £15m lower than this time last year, while beer is down £17m and burgers £6m.
“However, even unseasonable clouds have a silver lining. Shoppers sought refuge from the cooler weather and spent more on traditional comfort foods, with fresh and tinned soup sales up by 8 per cent and 16 per cent respectively.”
Analysts Nielsen also recorded sales growth of just 0.4 per cent across the sector in the last four weeks and more than £120m in missed sales due to the weather.
Mike Watkins, Nielsen's UK head of retailer and business insight, said: "It's clear that promotions, events and the vagaries of the weather have a big impact on supermarket sales."