The St. Paul Pioneer Press reported Monday that three customers had filed the two lawsuits in the U.S. District Court, seeking class action status to include other customers.
The company said last week that credit and debit card data for as many as 40 million customers had been breached.
One of the two lawsuits in Minneapolis focuses on the store's alleged neglect.
"Defendants indicated that it began investigating the incident 'as soon as (they) learned of it' but it did not contemporaneously disclose the breach to plaintiff and putative class members," wrote attorney Gregory McEwen, who represents Target customer Sarah Horton.
"Defendants have not made efforts to directly notify individuals whose information was compromised," the suit says.
The other lawsuit says Target's security system did not do the job.
"In one of the largest-ever commercial breaches of private information, Target failed to secure the payment information of its customers over the busy holiday shopping season," wrote attorney E. Michelle Drake, representing customers Theresa Burkstrand and Bryan Barth.
A Target spokesperson said the company does not comment on ongoing litigation.
Target said during the weekend it had notified millions of consumers of the breach that first began in late November, CNN reported.
JPMorgan Chase said it was limiting cash withdrawals at ATM machines to $100 a day and holding purchases to $300 a day for accounts that had been breached.
Hackers stole data that includes customer names, credit or debit card numbers and expiration dates.
Hackers also got their hands on the three-digit security code on the backs of the cards.
However, the personal identification numbers or PINS were not hacked, Target has said.