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. Doha denies the charges.
But despite strong mediation efforts by Kuwait and others, governments across the region say they may remain deadlocked for the foreseeable future.
"We can't have well off nations funding terrorism", Trump said in Wednesday's interview.
Tillerson arrived back in Doha after meeting Saudi Arabia's King Salman and Crown Prince Mohammed bin Salman 24 hours earlier.
On June 23, the four Arab states issued a list of 13 demands to end the rift with Doha, including closing of the Al-Jazeera television and cutting diplomatic ties with Iran.
Le Drian stressed that the crisis must be solved by the Gulf countries themselves, pointing out that France does not want to substitute anyone as it just wants to add its efforts to other countries efforts and support the Kuwaiti mediation. Staunchly defending its innocence against the allegations from the onset of the crisis, Qatar claimed the demands were deliberately created to be impossible for it to meet and rejected them, causing the current stalemate in negotiations.More news: If you wear disposable contact lenses - you should take heed of this
The report appeared to have been published by Swiss website The Local claimed the nations, excluding Libya, had called for Qatar to be stripped of the tournament but this has been rejected by Federation Internationale de Football Association. During Tillerson's recent trip, the United States did not offer any extraordinary initiatives or plans that would guarantee the settlement of the crisis.
On his previous visit on Tuesday, the USA and Qatar signed an agreement to combat terror funding, subsequently dismissed as "insufficient" by the Saudi-led states.
Tillerson signed a counterterror agreement with Qatar this week and called on all sides to negotiate, with his top aide telling the press in regard to all countries involved in the crisis: "No one's hands are clean here".
Al Jubeir stressed that there are basic principles that all countries, including Qatar, must abide by, including stopping support to terrorism..
Many analysts are sceptical about Washington's efforts to resolve the crisis between Qatar and Saudi Arabia and its allies, saying U.S. officials are not that willing to see the crisis settled soon because the perpetuation of the crisis would benefit Washington.
On Tuesday, speaking in Doha, Tillerson described Qatar as being "reasonable" in its dispute with the four states. All three are considered allies of many Western nations.