Ramtron's shares rose as much as 7 percent to a 10-month high of $2.61, well above Cypress's $2.48 per share offer.
"They are clearly going to test if there are other people beyond Cypress who want to buy them," Dougherty & Co analyst Charlie Anderson told Reuters.
Atmel Corp and Renesas Electronics Corp may take a look, he said.
Cypress's offer does not reflect the intrinsic value of the company and is not in the best interest of stockholders, Ramtron said in a statement.
Cypress offered to buy Ramtron last week, reviving a previously undisclosed bid it made last year.
"The board intends to consider the full range of available options including, but not limited to, the potential sale of the company or continuing with the company's current growth plans," Ramtron said.
Cypress has been invited to participate in the process, Ramtron said.
Analyst Anderson said this was a signal to Cypress to raise its offer.
Ramtron shares were up 5 percent at $2.57, while Cypress shares were flat at $13.55 on the Nasdaq.
(Reporting by Sruthi Ramakrishnan in Bangalore; Editing by Sriraj Kalluvila)
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