ICC's project officer in charge of corporate affairs, Jon Long, said Zimbabwean judge Ahmed Ebrahim, appointed to preside over the hearing, was willing to conduct full-day sessions to allow as much evidence as possible by witnesses representing Odumbe and the Kenya Cricket Association (KCA).
"The news of a verdict will take days and not weeks," Long said.
"The hearing will hopefully last for five days, after which Justice Ebrahim will fly to Zimbabwe to prepare his report, which he will submit to the ICC code of conduct commission," he added.
The report would require a seal of approval of the ICC executive board, which will forward the recommendations to the Kenyan authorities for appropriate action.
Odumbe, who has denied receiving payments from an Indian bookmaker to fix matches in 2002, faces a minimum two-year suspension or a possible life ban from the game.