Cambiasso, who will be 34 next month, is available on a free transfer after his contract with Inter Milan came to an end.
Lambert, who has only minimal funds with which to wheel and deal this summer, is attempting to persuade the Argentina international to come to England.
But he has a fight on his hands as Allardyce is prepared to offer big wages to land the defensive midfielder.
The Hammers hope to tempt Cambiasso to London on a two-year deal worth £50,000 a week - and the veteran star is considering the offer.
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