They told the court they needed access to Kenyatta's financial records, which they said might show that he had indirectly paid large sums of money to perpetrators of a wave of post-election violence that swept Kenya six years ago.
Kenyatta is charged with crimes against humanity related to the violence in early 2008 when 1,200 died and thousands were driven from their homes. He denies the charges.
"It is nearly two years since we asked for this material," said prosecution lawyer Ben Gumpert. "Of all the leads available to us, we have exhausted all reasonable prospects. But we are under a duty to pursue our investigations... The stones that remain to be turned are getting less and less promising," Gumpert said, explaining why access to Kenyatta's financial records could be crucial.
"We characterise the position of the government of Kenya as one of pure obstructionism," he said.
Prosecutors are asking judges to rule that Kenya is not meeting its obligations to the court and to adjourn the trial until it turns over further material in a case which has driven a wedge between African countries and the ICC's Western backers.
Kenyatta's lawyers, who want judges to dismiss the case, said the prosecution was attempting to abandon its case while pinning the blame for its failure on the Kenyan government.
The trial is important to the ICC, which has secured only one conviction and suffered a string of collapsed cases since it was set up 11 years ago. Kenya says the court risks destabilising east Africa if it presses on with the charges.
"CLIMATE OF FEAR"
Western countries, while keen to back the ICC, are also anxious to maintain relations with Kenya, seen as a key ally in the battle against militant Islamism in neighbouring Somalia.
ICC prosecutors say Kenyatta orchestrated the clashes, but their case has been weakened by the withdrawal of a string of witnesses since charges were first brought four years ago.
Presiding judge Kuniko Ozaki said the court would not rule on requests by the prosecution or the defence on Wednesday.
The trial was postponed for a fourth time last month when prosecutors said another witness had withdrawn, and requested more time to gather evidence. They say their witnesses have been blackmailed or intimidated into withdrawing.
In a January 31 court filing, prosecutors said a "climate of fear" had weakened their case and that judges should rule that Kenya was in breach of its obligation to help investigators.
But a lawyer for Kenyatta said the Kenyan government had been right to withhold the financial records since they had been requested directly by prosecutors, not by the court as a whole.
"The nature of the (January 31) filing is that this is an attempt to stop the case without admitting who has failed here," said defence lawyer Stephen Kay. "It is a blame-shifting exercise onto the government of Kenya."
The case grew more controversial throughout Africa after Kenyatta, the son of his country's founder, won a presidential election last year on a joint ticket with William Ruto, his deputy, who is on trial on similar but separate charges.
Following his victory, Kenyatta used his position as leader of East Africa's economic powerhouse to rally African Union allies in a diplomatic push to have the United Nations Security Council defer the case against him.
Although that was unsuccessful, the ICC's 122 member states did agree to change court rules to make it easier for heads of state facing charges to give evidence by video link.
Prosecutors say authorities have obstructed attempts to interview police officers, and have given investigators only limited access to phone records crucial to building their case.