Both sides have been lobbying hard in the US, where Lasmo investors are seen as more likely to hold out for a cash sweetener from Enterprise.
One analyst said: 'Shareholders in the US put more value on oil companies than in Britain and they will be more difficult to persuade. That is why Enterprise advisers are spending so much time out there.'
The battle for the hearts and minds of US shareholders comes as Hoare Govett, the stockbroker, gave a cool response to the bid. Analyst Colin Smith sees little synergy in the takeover and believes Enterprise can succeed only by raising its offer.
Graham Hearne, chairman and chief executive of Enterprise, is desperate to dispel such views, particularly in the US, where he flew immediately after launching the bid for Lasmo on 28 April.
But Enterprise's advisers had been there long before the announcement, laying the groundwork for what is the largest hostile offer in Britain since GEC and Siemens bid pounds 2bn for Plessey in 1989.
The rise in Lasmo's share price before the offer, and the Takeover Panel's demands for Enterprise to clarify its position, is thought to have stemmed from leaks in the US.
The recent downturn in Lasmo's financial fortunes led many UK shareholders to sell out to US investors, who believed that fears for the company's future were exaggerated. Lasmo also inherited a sizeable number of US shareholders after its disastrous pounds 1bn takeover of Ultramar. They include Scudder with 4 per cent, Capital Group, 4.7 per cent, and Brinson & Partners, 2 per cent.
An Enterprise adviser said: 'It is true that there is a sizeable audience in the US and we have been very keen to keep them in the picture. It is said that it will be difficult to shift their opinion behind us, but all I can tell you is that we have had a very good reception.'
A Lasmo adviser responded: 'Hearne and company have made no impact at all.'Reuse content