Hollywood Internet services firm J2 Global Inc. is pushing hard to close its unsolicited offer to buy the Boston backup storage provider Carbonite Inc. for $415 million.

J2 Chairman Richard Ressler on Tuesday took the company’s offer of $15 a share directly to Carbonite shareholders and told Carbonite’s new chief executive that j2 was not willing to wait to give the company more time to consider its buyout offer.

“We hope and expect that our tender offer will provide Carbonite stockholders a clear path to realizing full and fair value for their shares within a legally-defined timeframe,” Ressler said in a Dec. 23 letter to Mohamad Ali, who was named chief executive of Carbonite early this month.

The letter criticized the company for stalling, and for making a recent acquisition at a price j2 said was too high. In a letter received by j2 on Dec. 11, Carbonite apparently asked for additional time to consider the buyout offer.

J2’s takeover bid is scheduled to expire at 5 p.m. on Jan. 26. It is the company’s second run at Carbonite. When its 2012 offer of $10.50 a share was rejected as uncompetitive, j2 instead spent $21 million to purchase more than 9 percent of Carbonite’s outstanding shares, becoming one of its largest shareholders.

But J2, which had nearly $515 million in cash on its balance sheet at the close of the quarter ended Sept. 30, according to a regulatory filing, seeks the title of owner.

“Candidly, we were surprised that after more than two years since we first made an offer to acquire Carbonite, it would take Carbonite’s board up to another six weeks to act on an all-cash offer to acquire shares for $15,” Ressler said in the letter.

Shares of Carbonite rose 94 cents, or 7 percent on Wednesday to close at $14.54. J2 shares fell 46 cents, less than 1 percent, to closed at $63.12.