He added that the Foreign Office’s Crisis Response Hub is working to support at least 35 British nationals who have been directly affected by the earthquakes.
“We assess that the likelihood of large-scale British casualties remains low,” Mr Cleverly said.
Rescuers have been working in extremely cold temperatures searching for more survivors since the 7.8 magnitude earthquake struck the region on Monday morning.
At least 5,200 people have died across both Turkey and Syria – with the World Health Organisation (WHO) saying fatalities could reach as high as 20,000 in the coming days.
On the issue of aid, Mr Cleverly told MPs the Turkish government is requesting international assistance “on a scale that matches the enormity of the situation that they are facing”.
Updating the Commons on the UK response, he said: “Across the region inhabited by more than 12 million people, more than 6,000 buildings have collapsed. Electricity and gas infrastructure has been severely damaged.
“Many of the 3.5 million Syrian refugees hosted by Turkey reside in the affected provinces. Turkey’s outstanding disaster relief response capability has been severely tested by the sheer scale of this catastrophe.
“The Turkish government has declared a state of emergency and they are requesting international assistance on a scale that matches the enormity of the situation that they are facing.
“Turkey will lead the disaster relief response in the areas of Syria where it has the presence.”
Mr Cleverly also told MPs he has “already authorised the deployment of a medical assessment team” to respond to the earthquake, adding: “The further stages of requirement will evolve over time, we will of course work closely with our international partners to make sure we address that.”
Labour MP Feryal Clark told the Commons: “Like many British Kurdish-Turkish members of the community, my family woke up yesterday trying to find out whether our distant relatives, our relatives were safe in Turkey.
“We were very lucky that the majority of our family are safe.”