CKE Restaurants Inc. said Wednesday that it has received another, potentially better takeover bid than an earlier $928 million offer from a Boston private equity firm.

The Carpinteria parent of the Hardee’s and Carl’s Jr. fast food chains did not name the new bidder. Under the terms of its earlier merger agreement with Thomas H. Lee Partners, CKE can negotiate with a new suitor if the bid appears to be superior. It was allowed to seek alternative offers until Tuesday under a “go shop” clause.

Thomas H. Lee, which was among a group of investment firms that bought Dunkin’ Brands Inc. in 2006, in February offered $11.05 per share for CKE. The deal totals about $619 million in cash plus assumption of approximately $309 million in debt.

CKE, which operates or franchises 3,141 restaurants in 42 states, has until April 27 to negotiate with the second suitor. The company said that the new takeover bid does not yet include evidence of committed financing.

Shares were up 79 cents, or 7 percent, to $11.87 in midday trading on the New York Stock Exchange.