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A Manchester juror's facing contempt proceedings after claims he contacted a defendant from Bolton through Facebook.
It was during a multimillion pounds drugs trial causing it to collapse.
Attorney General Dominic Grieve will open the proceedings later in the High Court in London.
The Times said the case was against Joanne Fraill and an acquitted defendant, Jamie Sewart.
The case, to be heard by Lord Judge, the Lord Chief Justice, coincides with an appeal by another defendant in the trial who is challenging his conviction because of the alleged conversation, the newspaper reported.
The aborted trial involving Ms Fraill, 40, of Blackley, Manchester, and Ms Sewart, 34, of Bolton, took place in Manchester last year and involved multiple charges and defendants.
It was one of a series of four trials estimated to have cost £6 million and had run for 10 weeks.
One of the defendants, convicted drug dealer Gary Knox, is appealing against his conviction on the basis of alleged jury misconduct, with the appeal also to be heard by Lord Judge, the newspaper said.
Ms Sewart also faces contempt proceedings because she is alleged to have asked Ms Fraill for details of the jury's deliberations in relation to a charge they were still considering.
Knox, 35, was jailed for six years for conspiracy to commit misconduct in a public office, the newspaper said.
The court was told that he bought sensitive information on drug dealers from police in return for a £20,000 BMW and Premier League match tickets. A police officer, Phil Berry, 44, who received the gifts and admitted the same charge, was jailed for four years.