THE WALL STREET JOURNAL.

By SHARON TERLEP

The group trying to take Dell Inc. private is pressing the company to change how it counts shareholder votes to improve the odds of its buyout winning approval, and has offered to increase its bid by less than 1% as an incentive.

The move to renegotiate the vote rules midstream comes as the buyout group, Michael Dell and private-equity firm Silver Lake, have all but concluded they can't overcome entrenched opposition to their proposed $13.65-a-share offer from investor Carl Icahn and others unless the voting structure on the deal is adjusted, said one person familiar with their thinking.

In a letter revealed Wednesday, Mr. Dell and Silver Lake proposed that abstentions—which currently count as "no's"—don't count at all in the tally on their bid.

Dell's special board committee that negotiated the buyout said Wednesday that it is evaluating the offer, which also calls for raising the bid by 10 cents to $13.75 a share if the vote change is accepted.

The company Wednesday postponed for a second time, now until Aug. 2, a shareholder vote that had been scheduled on the bid. The first delay, last week, came after the original $24.4 billion buyout offer didn't appear to garner enough shareholder support to succeed.

Shares rose 3 cents to close at $12.91 Wednesday.

Mr. Dell made a personal plea in an open letter to shareholders late Wednesday, calling the revised offer in the best interests of the company and its shareholders. "The decision is now yours. I am at peace either way and I will honor your decision," he said.

The vote count issue stems from a concern at the root of the deal and some shareholder discontent with it: Mr. Dell's conflict in the transaction, as both buyer of investors' shares and a fiduciary of the company.

Mr. Dell is the company's founder, chairman, chief executive and its largest shareholder. To neutralize his influence in the sale process in an effort to make it fair to other shareholders, the company and buyout group agreed to certain terms in the voting process that are now proving too burdensome for the buyout group to overcome.

Under the agreement, the 16% of shares Mr. Dell controls don't count toward the vote, and non-votes count as "no's." Wednesday, the buyout group said 27% of eligible shares hadn't been voted; that amount combined with declared no's adds up to around 49% of shares that count in the buyout vote.

"The presumption that [nonvoting] shares should be treated as if they had voted against the transaction is patently unfair," the group said.

Heading into last week's scheduled shareholder meeting, the buyout group anticipated 10% to 15% of eligible voters wouldn't cast a ballot, rather than the 27% who ultimately didn't show, said one person familiar with the group.

"On Thursday morning, we were literally sitting there asking, 'Where are the votes?'" the person said.

After that, the buyout group made a big push to understand the makeup of those who didn't vote. The group feels confident that many of those who failed to vote aren't likely to oppose the offer, the person said. A number of the nonvoters, for instance, are investment firms that have policies against voting on corporate matters, the person said. Others were shareholders at the date counted for vote purposes but who no longer own shares.

Still, there is a risk that the turnout isn't the issue and they won't win the day unless they bump up their offer further.

People familiar with the deal described the issue over the vote count as difficult. On the one hand, changing the vote rules could look bad, raise questions about the process and invite shareholder lawsuits. At the same time, there are legitimate questions about how abstentions were shaping the tally, they said.

One person who works at an institutional investor that owns Dell shares said the revised offer means special committee members "are in a very tough position." The board panel may get sued if it stops counting non-votes as "no" votes, or it may get sued if it rejects the extra 10 cents a share, the investor representative said.

Lawyers not involved in the deal say the original structure would be fairly standard for a buyout by management; however, they also said there were enough protections in the deal for shareholders that the abstentions provision wasn't critical.

Richard Pzena, head of Pzena Investment Management LLC, the No. 20 shareholder in Dell as of the most recent data available, criticized the revised offer Wednesday and the board's handling of the situation.

"I can't believe 10 cents is worth changing the terms," Mr. Pzena said. "Obviously [Mr. Dell] can't win so he is trying to change the rules."

Danny Finocchiaro, a 29-year-old Boston-area resident and day trader, said Wednesday he was still on the fence on how to vote his approximately 31,000 shares of Dell. Mr. Finocchiaro said that if Mr. Dell raised the offer to $14 a share, such a boost would get the deal done, and the special committee should fight for that bigger increase.

The buyout group called the new offer its "best and final"—language it hadn't used earlier.

Mr. Icahn, Dell's second-largest shareholder behind Mr. Dell, has repeatedly criticized the bid.

"All would be swell at Dell if Michael and the board bid farewell," Mr. Icahn said in a tweet Wednesday morning.

—David Benoit, Kirsten Grind and Joann S. Lublin contributed to this article.

Write to Sharon Terlep at sharon.terlep@wsj.com

A version of this article appeared July 25, 2013, on page B1 in the U.S. edition of The Wall Street Journal, with the headline: Facing Defeat, Michael Dell Tries One Last Gambit.