Japan has not applied, citing concerns about the governance of the bank, but suggested that if it does join it may be after the deadline has passed.
"It's not a matter of setting a deadline. Our stance remains unchanged that we must cautiously watch how (China) guarantees fair governance at the AIIB," a government source told Reuters.
Japan's ambassador to Beijing, Masato Kitera, told the Financial Times that he believed the country would sign up by June 2015.
"The business community woke up late, but now they have mounted a big campaign for the AIIB which appears to be very effective," Kitera told the Financial Times.
If Japan does join the AIIB, the US will be the only large country left as a non-member.
When the UK announced its plans to join the bank on 12 March, the US told the Financial Times that this was part of a trend of "constant accommodation" to China.
China may reject Taiwan's application to join the Asia Infrastructure Investment Bank, because it does not recognise Taiwan as a separate country.
"We maintain that we should avoid the 'two Chinas' and 'one China, one Taiwan' situation," foreign ministry spokeswoman Hua Chunying told a briefing, Yahoo News reported.