Mobil Corp. today increased its offer for Conoco Inc. by more than $400 million, saying it would pay $105 a share in cash for 43.5 million shares and then exchange $85 in Mobil securities for the remaining shares.

The oil giant, the second biggest industrial company in the United States, said the total value of its bid is $8.2 billion, compared with a $7.3 billion cash and stock offer from E.I. du Pont de Nemours & Co. and a $4.08 billion all-cash attempt by Seagram Co. to acquire 51 percent of Conoco stock.

Meanwhile, Du Pont said today that Conoco shareholders have offered Du Pont about 35 million shares, roughly 41 percent of the company, while Seagram announced it has in hand 17 million shares, or 20 percent. Shareholders can withdraw those shares and offer them to another company until Aug. 4 in Du Pont's case and until July 31 in Seagram's offer.

The three-way quest to buy or control the nation's ninth biggest oil company is the most expensive and hotly contested in corporate history. Financial sources said that the bidding could go higher and that all three companies are likely to make changes in the details of their offers to try to make them more attractive to Conoco shareholders.

Conoco wants Du Pont to win the bidding war and has given the nation's biggest chemical company an option to purchase an additional 15.9 million shares at $87.50 a share. Mobil has gone to court to challenge the option because it gives Du Pont an advantage in the takeover battle.

"Mobil wants to show people they're still in the game," said one Wall Street trader who has a large holding of Conoco stock. Mobil's original offer was $90 a share in cash for 43.5 million shares and an exchange of securities -- preferred stock and a type of bond -- worth $90 a share for the rest. Today's new offer substantially boosts the cash portion of the offer, but knocks $5 a share off the securities portion.

One Wall Street source said Mobil "also probably is trying to get the price of the stock above the Seagram offer."

Seagram has said it will pay $92 a share for at least the first 44.35 million shares it receives. Conoco, whose trading was halted prior to the Mobil revised offer, rose $1.75 today to $88.50 a share. Seagram has secured enough bank credit to enable it to pay about $96 a share for Conoco stock.

Du Pont said it is willing to pay $95 a share for 34.44 million shares of Conoco and exchange 1.7 share of its own stock for the rest. But Du Pont Chairman Edward Jefferson said today that he will ask Du Pont's board at its Wednesday meeting to approve purchasing 38.7 million shares for cash.

Du Pont said it had its 35 million shares in hand as of midnight Friday, while Seagram's 17 million shares were on hand as of midnight last night. Wall Street sources said that on the so-called proration for Seagram -- last Wednesday -- the company had about 20 million shares, but about 3 million were withdrawn and tendered to Du Pont or Mobil. Sources say Mobil, whose offer is the richest but whose prospects because of antitrust problems may be the shakiest, has about 1.4 million shares on hand today.

The proration date is the day on which the bidding companies guarantee Conoco shareholders who have already shipped their stock what kind of remuneration they will receive. Mobil's proration date is Aug. 5.

The offers do not expire until next month, so stockholders can still send shares to any of the bidders, although after the proration date no shareholder knows for sure whether -- in the Seagram bid -- the stock will be accepted or in the other bids whether the type of payment desired (usually cash) will be assured.

After the proration date, companies treat the incoming shares on a first-come, first-served basis.

If, for example, Du Pont buys for $95 cash all the shares it has guaranteed it will buy, a stockholder in Conoco may get 1.7 shares of Du Pont stock (worth $77.56 today). Seagram might turn down a tendering stockholder if it has purchased all 51 percent of the stock it has said it will.

Conoco shareholders had not tendered enough shares to either Du Pont or Seagram by the proration dates to force those companies into accepting or rejecting shareholder preferences.