England's series with India has been marred by the fallout from the spat involving Anderson and Jadeja, which occurred during lunch on the second day of the opening Test at Trent Bridge.
Anderson faces a disciplinary hearing next Friday - the day after the conclusion of the third Test at Southampton - after being charged with a level three offence for allegedly "abusing and pushing" Jadeja, who has been fined 50 per cent of his match fee for his role in the incident.
Both captains have made remarks on the issue, England skipper Cook describing it as "mountain out of a molehill" on the eve of last week's second Test, while on Saturday Dhoni dubbed the decision to punish Jadeja as "hurtful".
The International Cricket Council (ICC) has been disappointed by those comments, and has asked all parties involved to stay silent and respect the judicial process.
An ICC statement read: "England captain Alastair Cook and, more recently, India captain Mahendra Singh Dhoni have made public comments relating to the alleged incident that took place during the first Test at Trent Bridge, the latter of which is critical of and undermines the ICC disciplinary process.
"The ICC would also like to reiterate its full support of the process followed and respect for the decision made by [match referee] David Boon. ICC Code of Conduct hearings of this nature are, generally speaking, difficult processes to work through, with detailed legal submissions, witness statements and oral witness testimony.
"We are satisfied that Mr Boon carefully and comprehensively considered the evidence and submissions presented to him at the hearing, which included conflicting evidence from both sides, prior to making his decision.
"In legal matters such as this, the ICC takes extremely seriously its duty as administrators of the game. Therefore, I request all stakeholders to respect the process which remains ongoing and remind them of their duties to the integrity of the process and the sport so that we can focus on the game of cricket itself."