The move is the latest escalation in a long-running dispute between the two sides over union claims that Murphy wrongfully sacked four Unite members at its Irish subsidiary last year.
Union leaders are now vowing to target the company “on an international scale.”
Unite said its campaign “will feature protests and demonstrations targeting Murphy and its key business partners, its customers and lenders.”
Initial protests this week have focused on Mott MacDonald who are supporting the joint venture between Bouygues and Murphy on the Lower Thames Crossing.
The union also staged a demonstration on Wednesday at a meeting of the British Tunnelling Society which counts Murphy among its members.
Unite warned: “The protests and ensuing disruption to clients and contractors associated with the company will further intensify.”
A Murphy spokesman said: “We categorically refute Unite’s misleading claims concerning the nature of this case.
“The company has always indicated that it is open to resolving this matter amicably and sensibly.
“We remain committed to maintaining an open dialogue with all parties around this issue to try to draw a line under this matter. This matter is currently subject to legal proceedings in Ireland.”