NASDAQ, the New York exchange that last week bid £2.4 billion for the London Stock Exchange, has hinted that it could offer its own shares or even increase the cash on offer to get the LSE board on its side.

However, the London market is continuing to reject the overtures and said that it would not meet Nasdaq even to discuss a higher offer.

Meanwhile, the New York Stock Exchange, another possible suitor, is thought to be putting together its own offer, although the NYSE refused to comment.

Nasdaq said last night it would press for an early meeting with the LSE board and that no options, including a shares offer and a higher bid, had been ruled out.

However, it emerged over the weekend that a phone call that had been arranged between Bob Greifeld, its chief executive, and Clara Furse, his counterpart in London, a week ago was not taken by Ms Furse.

The LSE takes the view, sources indicated, that the 950p on offer from Nasdaq was so far behind the exchange’s true value that no talks are necessary. Instead, the exchange is to press ahead with returning £510 million to shareholders, a pledge made last year but delayed by the auction for the company.

Today the Competition Commission will clear yet another possible suitor to bid. But the sharp rise in the LSE share price since the auction began in December 2004 means that there is little chance of a cash bid from Euronext, the continental exchanges operator, although some sort of agreed all-shares deal has not been ruled out.

The LSE share price has soared from below £4 in 2004 to close at 880p on Friday, before news of the Nasdaq approach broke, and a further surge is expected when the market opens this morning.

The LSE is encouraged in its stubborn approach by the high valuations placed on exchanges elsewhere, especially the US. Sources indicated privately that this had the full support of its main shareholders. “They [Nasdaq] are going to have to sharpen their pencils a lot more,” said one source close to the LSE.

By refusing to talk to Nasdaq, the LSE is also giving the NYSE more time to put together its own offer.

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