BAe's management, led by Dick Evans, the chief executive, is understood to be keen to top last week's blockbuster pounds 835m bid by GEC. In a series of meetings this weekend and over the next few days, BAe is hoping to convince shareholders of the merits of a new offer which would have to be around pounds 22 or pounds 23 a share, compared with pounds 21.50 from GEC.
But many BAe shareholders believe an offer this high would only damage the company. "Does a bid by BAe make sense at these levels? I don't think so," said one of the company's largest shareholders. "The bid put in by GEC is so high that this is probably the end game."
The shareholder added: "I would rather BAe did not get into a battle for macho reasons, to maintain its independence or anything like that."
The company would have a hard job convincing shareholders that it could gain the earnings benefits from buying VSEL at a price higher than the GEC offer. Instead it is likely to concentrate on persuading them of the strategic advantages of the takeover.
One of the main benefits of buying VSEL would be that the financial strength BAe would gain could make it invulnerable to any future takeover attempt by GEC. But this is not a factor that holds much importance for BAe shareholders, who care primarily about the company's earnings prospects.
The City was surprised by GEC's offer last Thursday because it was so far above the pounds 18.30-a-share bid by BAe. Observers believe GEC is indicating that it intends to top any bid put in by BAe. "BAe might be able to pay up to pounds 25 per share without too much dilution to its own shares," said Sandy Morris, engineering analyst at NatWest Markets.
"But the problem is that it is not as strategically important to BAe as it is to GEC, which needs to retain its dominance in naval contracting. This means GEC is always likely to offer more."
Thanks to its pounds 2bn cash pile, GEC's financial resources greatly exceed those of BAe.
A further problem is that BAe is offering its own shares for VSEL shares. If the City regards its bid as unrealistic, its shares will fall in value, reducing the value of its bid.
The stock market appeared to give a thumbsdown to any higher offer by the company last week when it marked BAe shares down 14p to 527p following the announcement of the GEC bid.
However, many in the City believe that even if it loses, BAe's management has already gained considerable credibility from its conduct of the bid.
Until recently, many observers believed that BAe's independent survival depended on winning the battle for VSEL. That view has changed. "BAe is not a borderline case any more," said one analyst. "It has rebuilt its balance sheet and can get by without VSEL."
The company may now have to focus more closely on its core business of aircraft manufacturing. In particular, it is likely to turn its attention to international alliances and joint ventures with the aim of gaining a dominant position in military aircraft contracting.
This would help it to balance GEC's superior position in warship contracting, which includes ownership of the Yarrow shipyard. The purchase of VSEL would give GEC virtual control of all Royal Navy contracts placed in Britain.
Despite the recovery in BAe's fortunes, however, there remains the danger that GEC may eventually try to purchase the company.
The decision by Michael Heseltine, President of the Board of Trade, to allow GEC to bid for VSEL is being taken in the City as an indication that the Government is content to see an increasing degree of consolidation in the defence industry at the expense of competition.