Carl’s Jr. parent, CKE Restaurants Inc., said Friday that it had agreed to be acquired by a Boston private equity firm for about $619 million in cash, plus assumption of $309 in debt.

The Carpinteria parent of the Carl’s Jr. and Hardee’s fast-food chains said that under the deal, Thomas H. Lee Partners will pay CKE shareholders $11.05 in cash for each share they own. That’s a 24 percent premium to the stock’s closing price on Thursday.

Thomas H. Lee Partners, or THL, has experience in the food sector. In 2006, it was part of a consortium that bought Dunkin’ Brands Inc., parent of Dunkin’ Donuts and Baskin-Robbins.

“THL’s proven history of success as an investor and value-added partner to its portfolio companies, coupled with its deep financial expertise and experience in the consumer sector, will also benefit all of our stakeholders, including our franchisees and our employees,” said CKE Chief Executive Andrew F. Puzder in a press release.

CKE operates and franchises more than 3,000 restaurants worldwide, using the Carl’s Jr. name in the Western United States, Hardee’s in the Southeast and Midwest, and both names overseas. The company, which promotes a higher quality image for its food, has seen sales stagnate during the recession as it resisted lowering prices. The company has not yet reported fourth quarter earnings.

CKE said the agreement allows it to seek superior offers until April 6. The sale otherwise is expected to close in the second quarter, pending shareholder and regulatory approvals.

CKE shares were up $2.19, or more than 24 percent, to $11.10 in midday trading on the New York Stock Exchange.