Conciliation service ACAS met union leaders, refinery owner Total, and main contractor Jacobs on Saturday, but no resolution had been agreed by this morning (Tuesday).
The dispute centres on Jacobs’ decision to award a subcontract to expand the refinery to Italian firm IREM, which brought in its own foreign labour.
The Lindsey strikers are being backed by 500 protesting workers at Stanlow refinery in Cheshire, while contractors in Fife have voted to continue their support through to Friday. On Monday, workers at Sellafield and Heysham nuclear power stations also came out in a show of solidarity.
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The strike action has been condemned by Prime Minister Gordon Brown and Business Secretary Lord Mandelson.
Strikers have used the slogan вЂBritish jobs fror British workers’ which Brown coined at the Labour conference 18 months ago.
But employment relations minister Pat McFadden said what Brown had meant was “to see the British workforce equipped for the jobs and skills of the future."
Lord Mandelson said: “It would be a huge mistake to retreat from a policy where, within the rules, UK companies can operate in Europe and European companies can operate here.
"Protectionism would be a sure-fire way of turning recession into depression."
However, environment secretary Hilary Benn said UK workers were "entitled" to answers in cases where they were not being used by foreign firms.
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