Sunday's devastating tsunami waves swept over Sri Lanka, killing at least 11,000 people there and leaving hundreds of thousands homeless.
Under the ICC's Future Tours Programme, the only stated reasons for countries to cancel a tour are the safety and security of its players or a direct instruction from a national government.
Failure to comply could lead to either a fine and even, possibly, suspension from international cricket.
Sri Lanka Cricket board president Mohan de Silva has been quoted as saying that his players would rather not play in New Zealand at this time and, while individually they all had a choice, their board as a whole had no option under the rules.
But ICC spokesman Brendan McClements told AFP in London via a telephone call on Tuesday: "There is always the option for a country to postpone a tour with the agreement of the host country. This is what happened earlier this year when Australia postponed its Test series in Zimbabwe, which had already been postponed from 2001."
"The other option is to talk to the ICC but no-one from Sri Lanka has spoken to us as yet," he added.
In the meantime Sri Lankan and New Zealand officials appeared to have reached a compromise solution which saw Wednesday's second one-day international due to be played in Napier postponed.
New Zealand Cricket chief executive Martin Snedden agreed to the Sri Lankan board's request for a five-day period of national mourning.
No players or managers lost immediate family in the tidal waves, but Sri Lanka manager Brendon Kuruppu, a former Test opener, said the team had been deeply affected by the tragedy.
"You cannot really assess the situation right now, it's something you cannot really imagine," Kuruppu said.
"Cricket is not the thing in our minds right at the moment. Two players' mothers were injured by floodwaters Sunday while a relative of another player's wife had been killed," team officials said.
The delay in the tour means the second one-day international is now set to take place on January 11, with the tour resuming in Queenstown on Sunday.
Scheduled matches in Christchurch on January 4 and Wellington on January 8 will follow, with Napier to round out the five-match series on January 11.
A two Test series due to open in Hamilton on January 15 will likely be pushed back several days to accommodate a warmup match beforehand, New Zealand Cricket officials said.
New Zealand won the first one-day match in Auckland on Sunday by seven wickets.