An ICC statement said that Ponting would be allowed to continue to use the bat, which is reinforced by carbon graphite, but denied media reports that it had already been cleared.
The world governing body said the matter was being looked into by the Marylebone Cricket Club (MCC) and the ICC Cricket Committee, chaired by former Indian captain Sunil Gavaskar.
"The ICC is growing increasingly concerned with issues surrounding the way in which bats are manufactured and 'enhanced'," chief executive Malcolm Speed said in the statement.
"During the ICC Cricket World Cup 2003 there were a number of oversized bats detected and manufacturers were put on notice that it was expected that they would produce bats that comply with the laws at all times."
Ponting started using the Kookaburra bat in Australia late last year.
It may be recalled that in the late 1970s, former Australian fast bowler Dennis Lillee was banned from using an aluminium bat in a Test against England.
Speed added that the MCC, as "the guardian of the Laws of the game", was the right body to investigate the issue and rejected criticism that it should not be involved.
"The issues with the current bat may be resolved, but that does not change the fact that the last thing that the sport wants is that, at some point in the future, the equivalent of golf's titanium driver is introduced without a full and proper debate. The MCC must be allowed to properly consider any new design."
The statement also said the ICC was worried about the "corking" of bats a practice also linked to baseball where the inside of bats is replaced by cork to make them lighter and easier to swing.
Speed added: "There is a lot of responsibility resting on the bat manufacturers to ensure that the equipment that they produce conforms to the Laws of the game and we want them to meet this responsibility."