The claim was made by Martin Hawkins, one of the ICC's Anti-Corruption and Security Unit officers (ACSU) who gave evidence at the hearing.
Hawkins said he investigated Odumbe's telephone calls, travels and hotel accommodation between January and December 2002.
He told the inquiry that he interviewed the player in his hotel room in Johannesburg during the World Cup in South Africa over his friendship with the bookie, Jagdish Sodha and his trips to Mumbai.
"Odumbe said he had been introduced to Sodha as someone he would do business of solar lighting with, when he retired from cricket. Sodha would also provide medicine for his mother who had hypertension," Hawkins said.
"It was the year before the World Cup and it was unusual for a player to make four trips to Mumbai and be entertained by Sodha," he added.
Hawkins said his investigations had proved that the player had received money and had his accommodation in Mumbai paid for by his host.
But Odumbe's lawyer Ishan Kapila claimed the ICC investigators had infringed on his client's private life by giving detailed reports on his girlfriends and other personal affairs.
"The code of conduct is dramatically fraud in the context," said Kapila, in his cross-examination.
Another ICC investigator, Niranjan Singh Virk, told the inquiry that he had interviewed Sodha in 2000 when he was still working for the Central Bureau of Investigation (CBI) in India and at the time, he confessed he had been a bookmaker.
He said Sodha had never been charged or convicted for the offence.
Odumbe is expected to give evidence in his defence on Friday.