While some of Mr Blair's colleagues believe that this kind of "no-strike" regime could be a means of avoiding disruption, unions have made it clear that such a system would end in a bureaucratic quagmire, with every petty argument ending up in the hands of an arbiter.
The strong disagreement among senior figures in the labour movement comes at a time when unions are planning more strikes on London Underground, postal workers are scheduled to stage another day-long walkout on Friday, civil servants have voted for stoppages at the Benefits Agency, and firefighters in Derbyshire announced further disruption.
Mr Blair has now moved, however, to placate trade unionists and it is understood that those in the Labour Party who are considering plans for compulsory arbitration have been told to keep their views to themselves. His assurance will inevitably lead to Tory accusations that Labour has once more bowed to its "paymasters".
As the Labour leader held one of his regular but secretive "contact group" meetings with trade unionists, Ian Lang, President of the Board of Trade, confirmed a threat to lift the Royal Mail's monopoly on delivering letters for less than pounds 1 unless employees abandoned their plan for a strike on Friday.
Talks between management and postal workers' representatives continued at the conciliation service Acas last night, and the postal executive of the Communications Workers Union is due to assess progress today. The union originally intended to stage a 36-hour strike beginning on Friday but has since curtailed it to 24 hours as an "olive branch" - although it is believed that some CWU members objected to the longer stoppage.
Elsewhere in the public sector, the CPSA Whitehall union announced a vote for strike action among members in the Benefits Agency who are concerned about their safety when the Jobseekers' Allowance is introduced in the autumn. Staff are worried that the strict rules may provoke angry scenes, and the union is seeking to reintroduce security screens for protection. The narrow majority for strike action - 5,449 to 4,434 in a 40 per cent turnout - may make it difficult, however, for the union to press ahead with stoppages.
The London Underground dispute remains deadlocked, with two further stoppages planned within the next seven days - one on Thursday and another next Monday - and six more after that.
In Derbyshire, leaders of 800 firefighters announced another four strikes in protest at cuts in spending which the Fire Brigades Union believes will result in the loss of 80 jobs.Reuse content