"I have talked to Indian State Minister for External Affairs MJ Akbar to help rescue Bangladeshis if anybody is affected by the cyclone," State Minister for Foreign Affairs M Shahriar Alam said in a Facebook message.
MJ Akbar assured his Bangladesh counterpart of providing support like they did while conducting rescue operation in Yemen.
Shahriar said Bangladesh officials will go to the affected areas once Salalah airport resumes operation.
Cyclone Mekunu caused flash flooding that tore away some roads and submerged others in Salalah, the country's third-largest city, leaving drivers stranded, reports AP on Saturday.
Strong winds knocked over street lights and ripped off roofs. The cyclone also struck neighbouring Yemen on Saturday.
Fast-moving waters from the rain and storm surge flooded normally dry creek beds.
Tourist beaches were littered with debris and foam from the Arabian Sea.
Capt Tarek al-Shanfari of Royal Oman police said the 12-year-old girl had been hit in the head by a door torn off by the wind.
An Asian person died in a flooded valley and an Omani national in a 4x4 was killed when his vehicle was swept away, Shanfari said.
Shahriar Alam said
many Bangladesh nationals remain engaged in fishing in Sea.
Bangladesh mission is in touch with other missions and authorities in Oman to get updated information.
Shahriar Alam hoped that there would not be many casualties as they were given alarm beforehand.
He urged all to keep Bangladesh nationals living in Salalah in their prayers.
Yemeni officials also reported damage in the east of the country along the border with Oman, reports AP.
Rageh Bakrit, the governor of al-Mahra province, said strong winds had destroyed houses and taken out communication lines and water services.
He said there were no deaths in the province.
India's meteorological department described the cyclone as "extremely severe", with sustained winds of about 110mph (177km/h).
Parts of Salalah, a city of about 200,000 people, lost power as the cyclone made landfall.
The airport, which has been closed since Thursday, will reopen early on Sunday, Oman's civil aviation authority said.
Salalah's port is a key gateway for the country.
Salalah and the surrounding area received nearly 11in (278mm) of rain, nearly three times its annual amount.