Story highlights More cases in Washington and in Oregon, bringing total up to 37

Chipotle says it is doing a deep cleaning and full sanitizing of its restaurants in the area

Cases are being investigated in connection with eight Chipotle restaurants

(CNN) At least 35 E. coli cases in Washington and Oregon have been linked to the Mexican-themed restaurant chain Chipotle, health officials said, prompting the company to temporarily close dozens of restaurants.

Two other E. coli cases in Washington were not definitively connected to Chipotle, the Washington State Department of Health said

The restaurant chain has responded by temporarily closing 43 restaurants in the two states affected, "even though only eight restaurants have drawn concern," the company said in a statement Tuesday.

The company is also deep cleaning and sanitizing all of its restaurants in that region, as well as conducting environmental and food testing in those locations, as well as testing in its distribution centers.

"The safety of our customers and integrity of our food supply has always been our highest priority," Steve Ells, chairman and co-CEO of Chipotle, said in a statement. "We work with a number of very fresh ingredients in order to serve our customers the highest-quality, best-tasting food we can. If there are opportunities to do better, we will push ourselves to find them and enhance our already high standards for food safety. Our deepest sympathies go out to those who have been affected by this situation and it is our greatest priority to ensure the safety of all of the food we serve and maintain our customers' confidence in eating at Chipotle."