J2 Global Inc. has been on a shopping spree, acquiring 10 companies in the last year.

Now, the Hollywood Internet services firm is hoping No. 11 will come from a previously spurned offer.

Carbonite, a backup storage provider for small and midsize businesses, on Wednesday said that it received an unsolicited takeover bid from J2 at $15 a share, a cash offer of just above $400 million.

That sent the Boston company’s shares up 24 percent to close at $14.44. The news also caused J2’s share price to jump 6 percent to close at $60.33 for the week ended Dec. 3, making it one of the biggest gainers on the LABJ Stock Index. (See page 28.)

A Carbonite spokeswoman said directors would soon begin reviewing the proposal and declined to comment further.

Representatives of J2 could not be reached for comment.

This is J2’s second run at Carbonite. When its 2012 offer of $10.50 a share was rebuffed as uncompetitive, J2 instead spent $21 million to purchase 9.9 percent of Carbonite’s outstanding shares, becoming its largest stockholder after Menlo Ventures.

But J2, which had nearly $515 million in cash on its balance sheet at the close of the quarter ended Sept. 30, still wants the title of owner.

In a Dec. 2 letter filed with the Securities and Exchange Commission, J2 Chairman Richard S. Ressler said Carbonite had yet to deliver the value Carbonite said it expected to create at the time of the first bid.

“We made clear that after more than two years as a patient stockholder, we believed it was time to see Carbonite take a strategic course that will maximize stockholder value,” he said. “Given that has not materialized, we believe the time has come for Carbonite to take affirmative steps to create that value.”

Shyam Patil, a senior analyst at Wedbush Securities in downtown Los Angeles, said J2’s second offer would be tough to turn down. At $15, the bid was a 27 percent premium to the $11.77 Carbonite shares closed at prior to the offer.

If the acquisition goes through, he said, Carbonite could increase J2’s earnings by about 20 percent. J2 said in November that it expected to hit upward of $600 million in revenue this year.

Carbonite’s online backup service would easily integrate with J2’s main business: its cloud service offerings. But if the acquisition is completed, J2 might not keep both of Carbonite’s business- and consumer-facing products, said Patil. The consumer side of Carbonite’s business isn’t profitable.

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