Colonel Mohammad Anisuzzaman, a Bangladeshi border official, said on Saturday: "Because of what is happening in the Bay of Bengal we have taken precautionary measures along the border.
"We have intensified the presence of our troops there and they are on high alert," he said.
Bangladeshi border officials had also alerted people living in the border area to leave their homes when the situation warranted.
Bangladesh sent a diplomatic team to Yangon in an attempt to resolve the issue through negotiations.
But a foreign ministry official said on Saturday: "The meeting ended without any resolution, but we have appraised our claim on the territory to the Myanmar authorities."
The two sides have been engaged in long-standing talks to agree on their maritime border in the Bay of Bengal, in which huge reserves of natural gas have been found.
When Myanmar recently sent naval vessels to support the Daewoo drilling operations, Bangladesh immediately deployed four warships to the area.
The next round of talks on the maritime boundaries are scheduled for November 16-17 in Dhaka, the Bangladeshi capital.