Thousands more are unable to cross the border, according to the migration office, or have found temporary shelter in Sittwe, the capital of the restive state of Rakhine in western Myanmar.
Most of those making the journey are women, children and elderly people.
The deadly fighting — between Myanmar’s security forces and a militant group known as the Arakan Rohingya Salvation Army — began when militants attacked army and police outposts near the border on Friday, prompting a swift crackdown by Myanmar’s government.
International human rights groups called the crackdown far-reaching and fear possible abuses against the Rohingya minority, who have long faced repression in Myanmar.
Those who crossed into Bangladesh this week have limited access to relief aid at makeshift camps. The country already hosts about 400,000 Rohingya who have fled Myanmar in recent years. The border between the two countries is officially closed.
In a statement released on Wednesday, William Swing, the migration office’s director general, urged international aid for those seeking refuge in Bangladesh, and called for the violence in Rakhine to end.
His remarks echoed calls earlier this week from Zeid Ra’ad al-Hussein, the United Nations High Commissioner for Human Rights, who condemned the violence in western Myanmar and said the government should “issue clear instructions to security forces to refrain from using disproportionate force.”
At the United Nations on Wednesday, Ambassador Matthew Rycroft of Britain called a Security Council meeting about Myanmar because of the violence.
“There is a threat to international peace and security, and it is right that the Security Council should take time today to be briefed on that and consider whether there is more that we should be doing,” he said before entering the meeting.
Asked what could be done, he said, “I doubt that there will be unanimity to do anything, as there are certain countries on the Council that tend to resist anything else, but I think it’s an important moment to take stock.”
On Wednesday in Yangon, hundreds of Buddhist nationalists gathered to support a harsher crackdown on the Rohingya.
The protesters urged security forces to assert control over Rakhine State and denounced an international advisory commission report issued last week that called for urgent government action to protect the rights of Muslims.
The commission, led by Kofi Annan, the former United Nations secretary general, urged Myanmar’s government to extend citizenship status for the Rohingya and allow them freedom of movement.
Mr. Annan said the commission’s proposal was intended to “trace a path to lasting peace, development and respect for the rule of law.”
“Violence will not bring lasting solutions to the acute problems that afflict Rakhine State,” he said.