"Our team found a corpse this morning and the firefighters have confirmed the death of this person," added a spokesperson for Eurotunnel.
The migrant, a man of Sudanese origin believed to be aged between 25 and 30, was hit by a truck that was leaving a cross-Channel ferry, the police source said.
Read: Illegal migration clearly linked with terror threat: Hungary PM
The latest fatality brings the number of migrants who died near the Channel Tunnel terminal site to nine since June.
"Everything happened overnight, and at 6am, the police still have quite a lot of work to do," said the police source, adding that "between 500 and 1,000 migrants" were still around the tunnel site.
The overnight attempts at storming the Eurotunnel terminal came just hours after some 2,000 migrants sought to enter the site on Tuesday, in what was described as the "biggest incursion effort in the past month and a half".
Security at the Calais port was stepped up in mid-June, driving migrants who previously tried to stow away on trucks that take ferries across the Channel to try their luck smuggling through the undersea tunnel.
A week ago, a Pakistani migrant died after being badly burnt while attempting to enter the Channel Tunnel in Calais in a bid to reach Britain.
“One of three migrants who were involved in the accident last week has died,” the northern Pas-de-Calais prefecture told AFP in a statement.
According to a migrant solidarity blog, the dead man was a 23-year-old Pakistani who succumbed on Friday to injuries he received on the night of July 13.
The three migrants had tried to board a ferry before entering the tunnel where they were hit by an electric charge.
The previous week, another migrant died in the Channel Tunnel.
In all, at least four people have died in and around the tunnel entrance in recent weeks, showing the risks incurred by migrants desperate to reach Britain.
Thousands of migrants are camped out around the port in the northern city of Calais, in the hope of climbing aboard lorries travelling to Britain on ferries or entering the nearby Channel Tunnel.
The migrants, whose presence has long caused friction between London and Paris, sometimes go to dramatic lengths to smuggle themselves into Britain, and have even been recorded trying to swim across the Channel.
In recent weeks, traffic through the Channel Tunnel has been repeatedly disrupted by protesting French sailors, as well as attempts by migrants to smuggle through the undersea passage.
Read: Increase in migration
The Eurotunnel company distributes pamphlets in nine languages near the tunnel to warn migrants of the risks involved in attempting to cross the Channel illegally.
Eurotunnel, which manages the Channel Tunnel and its vehicle shuttle services, also owns three ships that operate under the brand name MyFerryLink, which it bought from bankrupt French carrier SeaFrance in 2012.
SeaFrance crew members briefly blocked Calais port again on Monday angered by plans to sell two of the ships to rival ferry company DFDS.
COMMENTS
Comments are moderated and generally will be posted if they are on-topic and not abusive.
For more information, please see our Comments FAQ