They say, given the situation inside Myanmar, the Rohingya refugees' stay in Bangladesh may be a prolonged one or eventually be permanent unless Bangladesh authorities can take immediate measures to send them back.
Mahbubur Rahman, mayor of Cox's Bazar Municipality, said an estimated 10,000 newly arrived Rohingyas have already taken shelter in the beach town, while many others have moved to Chittagong, Bandarban, Rangamati and Khagrhachharhi.
Additional District Magistrate Khaled Mahmud said the authorities started working to "stop the Rohingyas from getting mixed up with Bangladeshis."
He also said they would strengthen monitoring on Sunday to stop the Rohingyas from spreading to other parts of the country.
According to the UN, around 290,000 Rohingyas had entered Bangladesh since Aug 25, when the Myanmar Army launched a counter-offensive in response to attacks on police posts and an army base in Rakhine.
The authorities in Cox's Bazar have strengthened security following the new wave of Rohingya influx.
Security personnel were seen patrolling the streets in the district, especially at Teknaf Upazila near the border with Myanmar on Saturday.
Many of them were seen travelling towards the beach town of Cox's Bazar by transports on Saturday.
Locals said the Rohingyas, who have money, do not face any problem to travel to any part of Bangladesh.
He said the Rohingyas were already staying in his municipality were sheltering the recently arrived refugees. "That's why we can't identify them easily."
Salima Khatun, a Rohingya woman from Akiyab in Rakhine, said the widow of one of her relatives killed in the recent violence in the Myanmar state moved to Cox's Bazar town.
Sharif Hossain, a Rohingya man from Maungdaw in Myanmar, was walking towards Kutupalong refugee camp in Teknaf from Ukhia.
He said he could have managed transports to Cox's Bazar if he had enough money.