Former captain Odumbe was banned for five years by the Kenya Cricket Association (KCA) in August for having inappropriate contact with a bookmaker.
Odumbe's estranged wife Katherine Maloney alleged during the tribunal that up to six other players were also involved.
The ICC's investigators are to talk to 10 players who represented Kenya between 1999 and 2000.
"I can't go into details of the interview. All I can say is that the ICC investigators started investigating myclients which they will do throughout this week," Allan Kosgey, representing the players, told Reuters on Monday.
"They are interviewing two players per day and they should dispense with the exercise by the end of the week," he said.
"I must add that the players and I are happy and satisfied with the line of interview. We are co-operating with how they are going about their business."
ICC investigators came to Nairobi for the exercise last November but the majority of players declined to appear, citing short notice.