The issue came about after the BCCI removed Bangladesh fast bowler Mustafizur Rahman from IPL side Kolkata Knight Riders for unspecified reasons, although there had been political tension between the countries.
The ICC then replaced Bangladesh with Scotland after the Asian country's government refused permission for them to travel and play in India, citing security concerns.
Bangladesh's decision to not travel to India strained the relationship between the BCB and the BCCI but Shakib is convinced it will be improved.
"I think things will get better," Shakib said on the sidelines of a EUT20 Belgium event in Mumbai. "The effort should be made (to improve relationship between BCCI and BCB).
"I think one or two series if Bangladesh visit to India or India go to Bangladesh; like I heard there's a series that is likely to happen in August or September.
"When that happens, I think the ice will break, and things will get better."
Shakib has also welcomed the decision of former international teammate Tamim Iqbal to become BCB president.
"I think he will have a long-term plan if he becomes president," Shakib said. "Hopefully Bangladesh cricket will get benefit from him."
Tawhid Hridoy expressed his delight after playing what he described as an "impact innings" to help Bangladesh beat New Zealand by six wicket in the first of three T20Is.
Captain Mehidy Hasan Miraz praised the "extraordinary" performance of seamer Mustafizur Rahman after Bangladesh sealed a 2-1 ODI series victory at home against New Zealand.
Nahid Rana took his second career five-wicket haul to help Bangladesh beat New Zealand by six wickets in Mirpur and level the three-match ODI series at 1-1.
New Zealand have arrived in Bangladesh for a proper six-match white-ball series without many of their regulars who are tied up in the Indian Premier League.
