Lawyers for Deputy Prime Minister Uhuru Kenyatta and Cabinet secretary Francis Muthaura told judges at a pretrial hearing that prosecutors have not disclosed key evidence ahead of the trial that is scheduled to start April 11.
They argue that the identities of several witnesses are still being withheld and large chunks of evidence have been redacted by prosecutors, meaning they cannot properly prepare for trial.
"The extent of the redactions, in names and substance, render the proposed date for trial entirely untenable," said Gillian Higgins, one of Kenyatta's attorneys.
Prosecutors said they are working within court rules and deadlines for disclosure set by the judges.
Judges made no immediate ruling.
Kenyatta and Muthaura are charged with crimes against humanity for alleged involvement in the murder, forcible deportation, persecution and rape of supporters of Prime Minister Raila Odinga in the aftermath of the 2007 vote.
The violence left more than 1,000 people dead.
Two other suspects, former Education Minister William Ruto and broadcaster Joshua Arap Sang, are charged with similar offenses and face a separate trial.
At a second pretrial hearing, lawyers for Sang and Ruto also argued for a delay in their clients' trial, arguing that the start date of April 10 would not give them enough time to prepare.
Kenyatta, who is the son of Kenya's founding father, Jomo Kenyatta, is running on a joint ticket with Ruto in next month's presidential election.
While Sang and Muthaura traveled to The Hague for their hearings, Kenyatta and Ruto remained in Kenya and participated in the hearings via a video link.
Also Thursday, Presiding Judge Kuniko Ozaki said that she wants two sets of judges to hear the cases. Currently, the same three-judge panel is due to sit in both trials.
Explaining how they will run the country while facing a trial at the ICC that could last for years, and how Kenya's relationship with the west will be affected if they win, has become the main issue for Kenyatta and Ruto in their campaigns throughout the country.
The top U.S. State Department official for Africa, Johnnie Carson, appeared last week to warn Kenyans against voting for Kenyatta. Shortly afterward, France and Switzerland said they would have only essential contact with Kenya's top leadership if Kenyatta wins the presidency.
Other presidential contenders have taken advantage of the ICC trial to discredit the alliance between the two suspects, in which Kenyatta is the presidential contender and Ruto is his running mate. Polls put Kenyatta as the second most popular presidential candidate.
On Monday during Kenya's first ever presidential debate, Prime Minister Raila Odinga, the front runner in the presidential race, said the country cannot be governed remotely. It will pose serious challenges to run a government via Skype from The Hague, Netherlands, Odinga quipped.
Kenyatta and Ruto have insisted that they will be able to rule Kenya if they win despite their trials.
A High Court in Kenya will rule Friday on whether Kenyatta and Ruto can run for election while facing trial at the ICC.
Associated Press writer Tom Odula in Nairobi, Kenya, contributed.