"France should be a facilitator in the mediation" led by Kuwait, Le Drian told reporters following talks with his Qatari counterpart Sheikh Mohammed bin Abdulrahman Al-Thani.
Le Drian flew in to Qatar at the start of a Gulf tour aimed at helping defuse the crisis pitting Qatar against Saudi Arabia, the United Arab Emirates (UAE), Bahrain and Egypt.
"France is very concerned by the sudden deterioration in relations between Qatar and many of its neighbours," said Le Drian, who also met with the emir, Sheikh Tamim bin Hamad Al-Thani.
"France is talking to all these countries to help in the search for a solution," he said, calling for "dialogue and calm" between the Arab states concerned.
Le Drian is to travel on to Saudi Arabia, Kuwait and the UAE, following a four-day mediation mission by US Secretary of State Rex Tillerson that ended on Thursday without any announcement of progress towards defusing mounting tensions in the Gulf.
Qatar's foreign minister welcomed France's support for mediation aimed at finding a solution "based on constructive dialogue... and respect of state sovereignty and international law".
"We look forward to France's support of the Kuwaiti mediation and American efforts," he said.
Riyadh and its allies imposed sanctions on Doha on June 5, including closing its only land border, denying Qatar access to their airspace and ordering their citizens back from the emirate.
They accuse Doha of supporting extremism and being too close to their arch-rival Iran.
Le Drian said France counted on "reinforcing cooperation with Qatar in the fight against terrorism, particularly in combatting terrorism financing".
His Qatari host said the fight against terrorism needed collective efforts and "cannot be shouldered by one state".
"Combatting terrorism also cannot be through practising political and intellectual terrorism against a state," Sheikh Mohammed said, in reference to measures taken against Doha by its neighbours.