The agrressive bidding war for Babcock & Wilcox Co. neared its conclusion yesterday as the two major contenders made what appeared to be their final, higher offers for the giant boiler manufacturer.

Unless J. Ray McDermott & Co. or United Technologies Corp. changes the terms of the offers made yesterday the winner should be apparent by early Thursday.

If McDermott does not have 2.5 million shares in hand by 5 p.m. today, it said it would drop out of the bidding and, presumably tender the shares it has acquired to United Technologies.

United, which started the bidding for Babcock & Wilcox last March at $42 a share, has said it would abandon its attempt to acquire control of the million shares in hand by 10 a.m. Thursday.

Yesterday, United Technologies raised the price it would pay for Babcock & Wilcoz to $58.50 a share. This fourth boost by United was a counter to last Friday's $60 offer from McDermott, a large, New Orleans based oil rig contractor.

Although McDermott's price was higher. United Technologies stands ready to buy all of the 12.2 million outstanding shares of Babcock & Wilcox, while McDermott will pay cash for no more than 4.3 million shares.

To counter yesterday morning's tender offer from United Technologies. McDermott boosted its offer price to $62.50 a share.

The fight for control of Babcock & Wilcox has been one of the most complex in history. Babcock & Wilcox has fought the takeover attemtp by United technologies in the courts and elsewhere, and the Justice Department is challenging the United Technologies bid on antitrust grounds.

United Technologies is a large conglomerate, based in Hartford, Conn., which among other things makes Pratt & Whitney aircraft engines, Sikorsky helicopters and recently acquired Otis Elevator.

Babcock & Wilcox is a major manufacturer of coal-fired and nuclear-fired boilers used to generate elec-electricity. Although it is based in tricity.

In fighting the takeover try, Babcock & Wilcox argued unsuccessfully last month that United Technologies is a major competitor because it manufactures gas-fired turbines, and a successful takeover therefore would violate antitrust laws.

The Justice Department is making the same case now before the same judge in U.S. District Court in Akron, Judge Leroy J. Contie, Contie said in mid-July that a merger of the two companies would not reduce competition in the utility boiler filed.

When McDermott jumped in the bidding fray earlier this month, offering $55 a share compared with the $48 offer United had put on the table earlier in the summer, Babcock & Wilcox endorsed the McDermott bid. Yesterday Babcock & Wilcox said it still "strongly recommends" the McDermott offer.

McDermott already controls about 10 per cent of Babcock and Wilcox stock. If it acquires 4.3 million more shares, it will own 45 per cent of Babcock & Wilcox stock.

United Technologies has said that its offer is conditioned on obtaining 6.4 million shares, which would give it 51 per cent, or control, of Babcock & Wilcox.

While United Technologies is prepared to pay cash for all 12.2 million shares - about $714 million - McDermott has said that it will pay cash for 4.3 million shares - about $269 million - and issue paper convertible into securities for any securities in excess of 4.3 million on a pro rata basis.

If McDermott is successful, the company has said it and Babcock & Wilcox would merge into a giant engergy equipment-manufacturing concern.

Analysts said that part of McDermott's motivation in the move is to avoid being taken over itself. McDermott has much cash in the bank and might prove a likely target for a cash-short conglomerate.

The bidding war has pushed the price of Babcock & Wilcox stock up. On March 1, Babcock & Wilcox closed at $30.50 a share. Yesterday the price was $57.25 a share, up 87.5 cent from Monday's close.