THE managers of the Rosyth Royal Dockyard in Fife have offered to undercut their rivals at Devonport near Plymouth by US$10 million in a last-ditch attempt to win the Trident nuclear submarine contract from the UK Defence Ministry. Babcock Thorn, which manages the dockyard, originally estimated the cost of extending the yard to fit Trident at $270 million, of which $120 million already has been spent. Devonport Management undercut this with a $170 million cost estimate for adapting its yard to handle Trident. However, Rosyth countered last month with a second proposal, estimating the cost of refitting at about $160 million. The Trident ballistic-missile submarines replace Polaris, but cannot be accommodated in existing dockyards. A spokesman for Babcock Thorn said the company's initial bid, which involved building a new facility, remained its preferred option, but it had been asked by the ministry to submit a second bid detailing the costs of refurbishing its existing docks instead. He refused to comment on details of the revised bid, but said: ''It will be no dearer than the Devonport option and it will very likely be considerably cheaper.''