New figures show that SuperValu and Tesco both held a 21.4% shares of the Irish grocery market in the 12 weeks to September 9.
But the figures from Kantar show that Dunnes is still the market leader with a share of 22.2% of the Irish grocery market.
Charlotte Scott from Kantar said that SuperValu returned to growth following two periods of negative growth and these 12 weeks represent its slowest share decline since September 2018.
Following a similar trend to the overall market, its shoppers have been making fewer, larger shops and taking home nearly one more item on average per trip.
Ms Scott also said that Tesco's sales decline softened to 0.5% as volume sales moved into growth, up 0.9% compared with last year.
"It might be too soon to call these improved performances from Tesco and SuperValu a fightback, but it will be welcome news given the recent success of Dunnes, Aldi and Lidl," she noted.
Dunnes achieved the second fastest growth during the past 12 weeks at 5.8%.
"Dunnes attracted an additional 38,000 shoppers compared with this time last year. If there was one negative in Dunnes' report card it would be that customers are visiting its stores less often – but that decline remains marginal at 0.3%," Charlotte Scott noted.
Meanwhile, Aldi increased its market share from 11.6% to 12.5% in the latest 12 weeks under review. As well as attracting new shoppers, Aldi also saw its customers getting bigger shops.
Kantar said that with each customer now picking up 19.7 items each visit, Aldi is clipping at the heels of Dunnes where the average is 20 - the highest in the market.
And while Lidl's growth slowed to 4.1%, it remains ahead of the market and has increased its share from 11.8% to 11.9%.
Today's figures from Kantar also showed that growth in the Irish grocery market remained at 2.7% during the 12 weeks to 8 September 2019, as shoppers stocked up on September staples.
Ms Scott said that when summer draws to a close people return to their normal routines and their shopping habits shift accordingly.
"Children are back at school and parents needing easy breakfast and packed-lunch options have boosted sales of cereals, yoghurts and bread by 14%, 10% and 6% respectively during the past four weeks, compared with the month before," she said.
She also noted that retailers have already been ramping up their festive activity with deals on seasonal chocolates including advent calendars.
Sales of chocolate boxes and gifts were 65% higher than the previous four weeks as people took the opportunity to get a head start on their shopping, she added.