Dell (Nasdaq: DELL), the No. 3 PC maker, seeking to battle Oracle (Nasdaq: ORCL), the No. 1 database developer, has bid as much as $2.34 billion for Quest Software (Nasdaq: QFST).

Buying Quest would further strengthen Dell's software and services tools for enterprises as it seeks to make more money on them than selling PCs and fit into CEO Michael S. Dell's strategy. It also would further upgrade Dell in the tools and services sector against Oracle (Nasdaq: ORCL), the No. 1 database and tools developer.

The Round Rock, Tex.-based manufacturer Monday upped its bid for Quest to $27.75 a share, or $2.34 billion, from last week's $25.50 a share,or $2.15 billion, for the Aliso Viejo, Calif., developer of software tools that enable enterprises to manage themselves better, investment bankers said.

The Dell bid topped an earlier offer from Insight Venture Partners in March of $23 a share. That accorded with the desire of Quest Software CEO Vincent C. Smith to take the company private under his management.

Insight, a New York-based private equity firm, recruited Vector Capital of San Francisco, another private equity firm, to top that with a new bid of $25.75, or about $2.17 billion.

In addition, Quest Software, which had agreed to Insight's first offer, upped the termination fee to $25 million from only $6.3 million.

Vector Thursday became the biggest shareholder of France's Technicolor (Paris: TCH) after a month-long battle with JPMorgan Chase (NYSE: JPM), the No. 1 U.S. bank. Vector has long taken stakes in technology companies either as long-term investor or to reshape it for future sale or spin-off.

Analyst Aaron Schwartz of Jefferies said he thought Dell's offer had been at the upper end, but now that the two private equity firms have joined forces, they could bid higher if necessary. Indeed, under the rules in which Quest agreed to Insight's first offer, any higher bid can be superceded by another that's made within three days.

Although Insight and Vector might come back with a bid of $28 a share, the likelihood of a competing bid is much lower now that Dell has topped its earlier ones, Schwartz said.

At the close of its first quarter on May 4, Dell reported cash and investments exceeding $17.2 billion.

The protracted tussle for Quest has rewarded its shareholders. They jumped more than 5 percent Monday to $27.61, up $1.39, up nearly 49 percent for the year, with most of the gain since Insight made its initial bid.

Dell shares fell 26 cents to $11.94 in Monday trading and are down more than 18 percent in 2012 and more than 27 percent for the past 52 weeks.

Oracle shares fell 38 cents to $27.62. They're up more than  percent in 2012 but down 11 percent for the past year.