Dyson, a former Sri Lanka coach, had been in talks with the Bangladesh Cricket Board to replace Dav Whatmore, who had refused to sign an extension to his contract in May.
"It's shocking for us. We came very close to appointing him," Gazi Ashraf Hossain, chairman of BCB's Cricket Operations Committee, told Reuters.
"We even discussed when and on which flight he would come to Dhaka to sign an agreement.
"Against this backdrop, his move to West Indies is surprising. It is a major setback for us."
The BCB moved to sign Dyson last month after their three shortlisted candidates, Johan Harmer, Jamie Siddons and Dave Houghton, all opted out.
Dyson, who played 30 tests and 29 one-dayers between 1977 and 1984, was named coach of the West Indies on Saturday.
He was in charge of Sri Lanka for almost two years from 2003 and succeeded compatriot David Moore as West Indies coach.
The BCB faces a race to appoint a coach before Bangladesh's tour of New Zealand in December, where they will play two tests and three one-day internationals.
They have set a deadline of October 30 to appoint a coach, although Ashraf said Dyson's decision to turn down the job had made the process all the more difficult.