The transfer of the men across a bridge over the Naf River separating the two countries began late morning, AFP reporters on the scene said.
"I'm happy," one of the men, who gave his name as Uzzal, told AFP in English. "Four months after, go back to Bangladesh, to the family, very happy."
In recent years tens of thousands of persecuted Rohingya Muslims from Myanmar and Bangladeshi economic migrants have fled on boats across the Bay of Bengal in search of better prospects, usually to Malaysia.
Since then, around 4,500 of them have returned to shore, but the UN estimates around 2,000 are still at sea.
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Nearly 1,000 have been taken to Myanmar's western Rakhine state, which borders Bangladesh, after being found by Myanmar's navy in two boats in the Bay of Bengal.
Neither nation initially showed a willingness to accept them and rights groups are concerned some could be pushed to the wrong side of the border.
But they now appear to have agreed on the nationality of some of the rescued migrants, who come from a first boatload of more than 200 found in late May.
After accepting 150 people back, a Bangladeshi border official said the complicated process of establishing where the remainder belong was ongoing.
"Only after the verification we can say how many or if any Bangladeshis are among these migrants," Lieutenant Colonel Khandaker Saiful Alam, head of Bangladesh border guards in Cox's Bazar district bordering Myanmar, told AFP.