Updated from 7:14 p.m. EDT

Two days after suggesting a sweetened bid to buy

McAfee.com

(MCAF)

,

Network Associates

(NET) - Get Report

on Monday upped that offer even further.

In a press release Monday after markets closed, Network Associates announced it was increasing its exchange offer to 0.90 of a share of Network Associates for every share of McAfee it does not already own. The offer is conditioned on McAfee not adopting any "structural defense, including the adoption of a shareholder rights plan," Network Associates said in the press release.

In a signal that some bargaining may be occurring between the two closely tied companies, the latest offer came on the same day an email suggesting a slightly lower 0.85 exchange was made public.

In an email Saturday, Network Associates CEO George Samenuk wrote to Frank Gill and Richard Schell, members of a McAfee board of directors special committee, that his company is prepared to pay 0.85 share of Network Associates for every share of McAfee that it does not already own.

He said that offer assumes he receives assurances from the special committee that it will favorably recommend the new offer and that the McAfee board or special committee would take no action to cause the offer conditions to fail, such as adopting a poison pill. The email was disclosed Monday in a filing with the

Securities and Exchange Commission

.

On July 1, Network Associates announced its third bid to buy 25% of McAfee that it doesn't already own. The company offered to pay 0.78 of a share of Network Associates for every outstanding share of McAfee common stock in a deal valued at $180.4 million based on June 28 closing prices.

That was the same sweetened exchange ratio previously offered by Network Associates on April 10 and was a slight premium over McAfee's June 28 closing price. But it is lower than the April 10 offer, worth $224 million, because Network Associates' stock price has fallen.

Network Associates abandoned its previous attempt to buy back McAfee, which it spun off during the dot-com boom to sell security software via the Internet, because it was forced to restate financial numbers. Network Associates is still being investigated by the SEC.

Based on Monday's closing prices, the .90 exchange ratio offers a premium of 55 cents a share to McAfee stockholders. But the total price still comes in at only $137.7 million.

Shares of Network Associates fell $1.25, or 8.9%, to close at $12.75. Shares remained flat in after-hours trading. McAfee shares fell 98 cents, or 8.2%, to close at $10.93. Shares bounced up to $11.50 in after-hours trading.