Insight Venture Partners not only raised its offer in the battle for Quest Software Inc. (QSFT), but the buyout firm also increased the stakes for the other party competing to purchase the maker of database-management software.

Specifically, Insight's offer to make Quest Software a private company is now valued at $2.17 billion, up from $1.94 billion in March, and includes a breakup fee nearly four times higher than what was provided in the firm's previous offer. Insight's new bid was also deemed superior by Quest's board.

It was unknown how the competing bidder--which hasn't been named officially but is computer maker Dell Inc. (DELL), according to a person familiar with the situation--would react to Insight's higher offer. A Dell spokesman declined to comment on anything related to Quest, but observers expected the company to stand down.

"Dell has walked away from bidding wars in the past," Wunderlich Securities analyst Brian Freed said. "They don't let pride get in the way of common sense."

The fight over Quest echoes a similar battle Dell waged for high-end data-storage maker 3Par Inc. in 2010. Dell eventually bowed out after Hewlett-Packard Co. (HPQ) bid $1.6 billion for the company, which makes rapidly accessible storage drives that allow corporations to run more efficient databases.

Insight's latest offer, which now includes fellow private-equity firm Vector Capital, values Quest at $25.75 a share, higher than the offer of $25.50 a share last week from Dell and above Insight's original proposal of $23 a share in March.

Quest shares on Wednesday recently slid a penny but, at $26.51, was still above the latest offer, suggesting some investors are still hoping for a counteroffer.

Dell can afford to pay more for the company, but it would be costly. Any new offer from the PC maker not only has to be higher than $25.75 a share, but it also has to account for the $25 million break-up fee, which is greater than the $6.3 million promised in Insight's earlier offer.

Plus, at the current price, Quest can be viewed as expensive. The latest offer values Quest at 15.6 times future earnings, which is a valuation nearly 50% higher than that of Microsoft Corp. (MSFT) and Oracle Corp. (ORCL). And Quest isn't seen as a fast-growing company, with sales projected to rise a modest 9% this year and 6% next year to $990 million.

Quest, which has more than 100,000 customers, provides software associated with infrastructure hardware, such as databases, servers, operating systems and virtual environments. The company has faced increased competition from companies that build enterprise systems for which it traditionally provided software, such as Oracle, which has been building its own tools to manage systems.

The Aliso Viejo, Calif., company has responded to the increased competition by making acquisitions. The extra business has grown revenue, but the costs surrounding the deals have contributed to Quest's earnings getting cut in half last year.

Quest Chairman and Chief Executive Vinny Smith has been a constant presence in most of the software developer's 25-year history and featured prominently in its latest go-private offer. Smith would keep most of his 34% stake in Quest under the deal with Insight and Vector, along with executive control of the company he helped lead through its 1999 initial public offering.

--Anupreeta Das contributed to this story

Write to Drew Fitzgerald at andrew.fitzgerald@dowjones.com

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