Several EU countries have upped security efforts in the wake of the suspected terror attack.
A tractor trailer slammed into a crowded Christmas market in Berlin on Monday night, killing 12 people and injuring 48 others. Witnesses said shoppers screamed and dropped packages and glasses of mulled wine as the truck ploughed into the market.
Officials are investigating the crash as an act of terrorism, according to a German intelligence official.
After a tip, Berlin police arrested a man on foot about a mile and a half away from the crash site. Authorities are working to determine whether he is linked to the case.
A 23-year-old Pakistani asylum seeker named as Naved B, who entered the country under a false name, was arrested on suspicion of murder.
However, there are reports in Germany that police now believe he was not the perpetrator and that the true attacker is "still armed, at large and can cause fresh damage."
According to security sources, authorities in the German capital had information in recent days about a possible attack against a Christmas market.
Speaking on Tuesday, Juncker said his thoughts were with the families of the victims, saying the news was made even more devastating "because they had gathered there to celebrate the pre-Christmas season, which unites many with peace."
Elsewhere, Parliament's GUE/NGL group leader Gabi Zimmer, said, "We condemn in the strongest terms the indiscriminate attack on the Berlin Christmas market yesterday.
"We express our deepest condolences to the families of the victims and wish the injured a speedy recovery. We thank the rescue teams for their brave efforts and bystanders at the market who were first to provide assistance to those in need."
"The people responsible for the attack must be brought to justice. Nothing justifies this cowardly violence. However, we must not be guided by fear, nor allow this atrocity to be exploited for political gain. It is important at this difficult time to stand firm for the values that define us such as solidarity, peace and tolerance."
"Grief for the Christmas massacre in Berlin, solidarity with Chancellor Angela Merkel and all of the German people," said Italian Prime Minister Bernard Gentiloni.
"The French share in the mourning of the Germans in the face of this tragedy that has hit all of Europe," French President François Hollande said.
The French Interior Ministry said it strengthened security measures at its Christmas markets.