The company beat its nearest rival, Zee Telefilms, by over $23 million, according to details of the bidding revealed by the Bangladesh Cricket Board (BCB).
"We will sign a contract with Nimbus within the next two weeks. All we need now is the government's seal of approval," BCB general secretary Mahbub Anam said on Friday.
The previous contract, with ESPN-Star Sports, was worth $11.75 million for a five year-period and expired in April this year.
Under the new marketing rights deal, Nimbus will telecast live all the Test and one-day matches earmarked for Bangladesh in the Future Tour Program (FTP) of world cricket's governing body, the International Cricket Council.
Bangladesh's FTP begins later this month with a five-match, one-day series against Zimbabwe at home.
According to the FTP, Bangladesh will play at least 21-23 Test matches and 59-61 one-day matches in the next six years.
Nimbus has already won the rights to telecast Indian cricket matches up to 2010. It paid 612 million dollars to secure those rights.
The company also announced it would launch a 24-hour cricket channel, which will be seen in millions of Indian homes via the third-party Star platform that reached 96 percent of Indian homes with cable.
Bangladesh's cricket officials hailed the deal.
"It's a great achievement for our cricket. It indicates Bangladesh cricket is on the upward move," Anam said.
"We'll invest the money to build cricketing infrastructure and tournaments in the country. It will be a big fillip to our cricketing development," he said.
Apart from telecasting international matches, the company will have to telecast a minimum 10 days of domestic matches under the deal, he added.