Opposition groups have claimed chlorine gas attacks on at least three towns left dozens of people injured last week.
"We have a few elements of information, but I do not have the proof," French President Francois Hollande told Europe 1 radio station.
"What I do know is what we have seen from this regime is the horrific methods it is capable of using and the rejection of any political transition."
French foreign minister Laurent Fabius told the radio station there were "indications, which have yet to be verified, that there have been recent chemical attacks".
A bomb attack in the Dahra Awad neighbourhood of Aleppo on Sunday
He said they were "much less significant than those in Damascus a few months ago but very deadly", and had taken place in the northwest of the country, near the Lebanese border.
There are conflicting accounts about one attack in the town of Kafr Zita, near Hama, earlier in April, with both Mr Assad's forces and the opposition accusing each other of being responsible.
Activists accused the regime of using lethal chlorine gas - first used at Ypres during the First World War - saying it caused "more than 100 cases of suffocation".
Videos posted on social media by opposition groups show men and children in a field hospital coughing and showing symptoms of suffocation.
Hollande says he has information about chemical attacks in Syria
Asked about the reports, a Foreign Office spokesperson said: "We need to establish the facts, but are seriously concerned and nothing would surprise us given the regime's history of using chemical weapons.''
The Syrian regime has handed over 80% of its declared chemical weapons stockpile for destruction under UN supervision.
It has until the end of June to destroy its chemical weapon stockpile if it wants to ward off the threat of US air strikes.
The agreement was reached after a chemical attack outside Damascus last August that killed hundreds of people.
Meanwhile, it was announced that presidential elections - in which Mr Assad will seek another term - will be held on June 3.
