The sources say the purpose may be to prevent the return of Rohingya Muslims fleeing the violence.
Bangladesh will on Wednesday formally lodge a protest against the laying of landmines so close to the border, the sources - who had direct knowledge of the situation but asked not to be named because of the sensitivity of the matter - told Reuters news agency.
Since the latest round of violence began in Myanmar's Rakhine state, at least 400 people have been killed, and nearly 125,000 Rohingya have fled to neighbouring Bangladesh, leading to a major humanitarian crisis.
"They are putting the landmines in their territory along the barbed-wire fence" between a series of border pillars, one of the sources told Reuters.
Both sources said Bangladesh learned about the landmines mainly through photographic evidence and informers.
"Our forces have also seen three to four groups working near the barbed wire fence, putting something into the ground," one of the sources said.
"We then confirmed with our informers that they were laying landmines."
The sources did not clarify if the groups were in uniform, but added that they were sure they were not Rohingya.
Reacting to the reports, Phone Tint, Rakhine's minister for border affairs, told Al Jazeera: "We did not do such a thing."
Manzurul Hassan Khan, a Bangladeshi border guard officer, told Reuters earlier that two blasts were heard on Tuesday on the Myanmar side.
Two similar blasts on Monday had already prompted speculation that Myanmar forces had laid landmines.
One boy had his left leg blown off on Tuesday near a border crossing before being brought to Bangladesh for treatment, while another boy suffered minor injuries, Khan said, adding that the blast could have been a mine explosion.
A Rohingya refugee who went to the site of the blast on Monday - on a footpath near where civilians fleeing violence are huddled in what is being described as "no man's land" on the border - filmed what appeared to be a mine: a metal disc about 10cm in diameter partially buried in the mud.
He said he believed there were two more such devices buried in the ground.
Two refugees also told Reuters they saw members of the Myanmar army around the site in the immediate period preceding the Monday blasts, which occurred at around 2:25pm local time (07:55 GMT).
Reuters was unable to independently verify that the planted devices were landmines and that there was any link to the Myanmar army.
Myanmar's army has not commented on the blasts near the border.
Zaw Htay, spokesperson for Myanmar's national leader, Aung San Suu Kyi, was not immediately available for comment.
On Monday, Htay told Reuters that clarification was needed.
"Where did it explode, who can go there and who laid those landmines. Who can surely say those mines were not laid by the terrorists?" he said.
The border pillars mentioned by the Dhaka-based sources demarcate the boundaries of the two countries, along which Myanmar has a portion of barbed wire fencing.
Most of the two countries' 217km-long border is porous.
"They are not doing anything on Bangladeshi soil," one of the sources said.
"But we have not seen such laying of landmines in the border before."
Myanmar, which was under military rule until recently, is one of the few countries that have not signed the 1997 UN Mine Ban Treaty.
The more than one million Rohingya in Myanmar are seen as illegal immigrants in the mainly Buddhist country.
They have been forced to live under apartheid-like restrictions on movement and citizenship.
The areas where Rohingya live, mainly in Rakhine, have been under a constant military crackdown, with reports of extrajudicial killings, rape, arson and torture by security forces - allegations the government has denied.
He warned of "regional destabilisation" if the violence continues.
Myanmar contends the security crackdowns are necessary to fight "terrorism".
In a statement issued by her office on Facebook, Aung San Suu Kyi said the government had "already started defending all the people in Rakhine in the best way possible" and warned against misinformation that could mar relations with other countries.
Al Jazeera's James Bays, reporting from the United Nations headquarters in New York, said: "There is a real concern from the UN about the humanitarian situation because of this human exodus and the sheer number of people crossing the border into Bangladesh."
The UN Security Council met last week to discuss the crisis, but there was no formal statement following the closed-door meeting.
Source: Al Jazeera and news agencies