Asda's US parent company Wal-Mart has called for the government to launch an investigation into Tesco's domination of the UK supermarket sector.
Asda and Tesco have a frosty relationship at present
US-based Wal-Mart, the world's largest retailer, told the Sunday Times that such a probe was vital now that Tesco's UK market share had hit a record 30.5%.
Tesco, the UK's largest supermarket group to Asda's number two, said the 30.5% figure was "misleading".
Wal-Mart's comments came as Asda unveiled £130m ($234m) in price cuts.
Tesco's 30.5% market share figure was published last week by independent market research company TNS. Asda's share was given as 16.7%.
Wal-Mart boss Lee Scott said the government had to investigate Tesco's continuing domination because it was so difficult for rivals to try to catch up.
"As you get over 30% and higher, I am sure there is a point where government is compelled to intervene, particular in the UK, where you have the planning laws that make it difficult to compete," said Mr Scott.
Tesco said the 30.5% claim was misleading because it did not factor in food sales from Marks & Spencer and other department stores.
It said that when these were taken into account, its share of the overall UK food market was just 16.7%.
Tesco also claimed that it was acting in the best interests of shoppers.
"Previous Competition Commission inquiries have found that the market - and Tesco - operates in the consumer interest," said Tesco's corporate and legal affairs director Lucy Neville-Rolfe.
"It is a competitive market. The consumer is the winner."
Asda is introducing more than £130m of price cuts as part of its 40th anniversary celebrations.
Earlier this month the Advertising Standards Authority ordered Asda to drop its claim to be "officially the UK's cheapest supermarket".
It followed a complaint by Tesco that Asda's claim was based on too limited and unrepresentative a survey.