The chain is planning an 820,000 sq ft distribution centre on a former air force base in California, managed by the March Joint Powers Authority, which controls civilian use of former miltary property.
But the lawsuits claim the authority did not study the effects on traffic and air quality that the centre will have.
Tesco insisted it was on track to open its US stores next year. "Challenges of this nature are apparently quite common in California," said a spokesman. "We have every reason to believe the decision by the authority will be upheld." There was no indication of when the cases would be resolved, he added.
Tesco has said it will start trading in the US next autumn. It has already recruited 100 people at its head office in El Segundo, California.