The “extensive” operation was launched after two Somali men travelling in a small vessel were rescued by Border Force and HM Coastguard near Harwich on Monday afternoon.
A third person, assumed to have been travelling with them from France, reportedly entered the sea in international waters.
However, search efforts made by Border Force and the RNLI and co-ordinated by the Maritime and Coastguard Agency concluded at 2pm on Tuesday.
The RNLI said lifeboats from Harwich and Walton and Frinton were involved in the rescue operation.
The Harwich lifeboat was launched just after 4pm on Monday and again at 11.35pm the same day.
The Home Office spokesperson thanked those who had helped with the emergency response. While investigations are ongoing, they said the incident was a “reminder of the extreme dangers of crossing the Channel in small boats”.
“We are determined to do everything we can to prevent people dying in the Channel,” the spokesperson added.
Enver Solomon, chief executive of the Refugee Council, said it is “not the first time” someone had gone missing crossing the channel “in search of safety”.
The Refugee Council boss urged the government to “fundamentally” change its “cruel and ineffective” approach to migration by opening up safe routes for asylum seekers.
“Every day, people are forced to flee their home through no fault of their own and we must do more to make the journey safer,” Mr Solomon said.
Local MP Sir Bernard Jenkin told Harwich the Harwich and Manningtree Standard that the “potential loss of life is extremely distressing”.
He also said that more needed to be done to tackle people smuggling gangs.
On Tuesday, the UK rescued or intercepted 145 asylum seekers in the Channel, while France prevented 137 from reaching England, according to the Home Office.