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"We have nothing to believe this is an issue for us at this moment," Lundgren told "Squawk Box," adding that securing customer data is much broader than one or two companies. "Everyone was on alert."
Against the backdrop of these attacks, there have been calls to upgrade the technology behind credit card purchases for Americans customers.
In an exclusive CNBC interview that aired Monday, Target Chairman and CEO Gregg Steinhafel said he supports the movement to a chip and PIN-number system called EMV technology that replaces vulnerable magnetic strips.
(Read more: 'I'm still shaken'by Target data breach, says CEO)
But switching the entire payment system will be difficult and expensive—and it likely won't happen for shoppers in the U.S. until the end of 2015.
Lundgren didn't say whether he supports that EMV system, but stressed: "The retailers, the banking industry, [and] the credit card industry should be working very closely together to figure out what is the right technology to protect the consumers … and then work around the solutions from there."
On Dec. 19, Target disclosed that as many 40 million credit and debit cards were compromised between Nov. 27 and Dec. 15 by malware installed on the company's point of sale registers.
(Read more: Target CEO defends delay to disclose data breach)
Target said Friday its continuing investigation found that at least 70 million customers' personal information was stolen from its database—including names, mailing addresses, telephone numbers, and email addresses. The retailer expects some overlap in the two data sets but does not have the exact numbers yet.
Also Friday, Neiman Marcus disclosed that some its customers' credit cards were compromised in a security breach.
"The Target breach can get us all moving" to safer credit card systems to protect consumers, said Steve Sadove, former chairman and CEO at Saks and current chairman of the National Retail Federation (NRF).
"We do need to move to some new technology," he continued. "It's going to require cooperation with the banks as well as the retailers."
"We're not the only stakeholders here," NRF president and CEO Matthew Shay emphasized. "We need financial institutions. … We [all] have an interest in protecting the consumer. That's [our] number one interest and it should be for the other stakeholders as well."