"I would like to express my support and our support for our ambassador and for all the embassy staff, who are doing a remarkable job in conditions which, as you know, are difficult," Colonna told LCI TV, adding that the letter had requested a change in envoy.
In December, Burkina Faso sent a letter requesting the departure of France's ambassador - a move the French government described as "not standard practice."
A French diplomatic source said that at this stage Paris was not planning to withdraw its envoy and still assessing the objective of Burkina's request.
Relations between France and Burkina, a former French colony in West Africa, have deteriorated after repeated military coups.
Protests by opponents of the French military presence have surged, partly linked to perceptions that France has not done enough to tackle an Islamist insurgency that has spread in recent years from neighboring Mali.
France retains some 400 special forces based in Burkina. It withdrew forces from Mali last year after the military junta there agreed a deal with Russia's private Wagner Group, which is staffed by Russian army veterans and prison convicts, to operate in the country.
"There was a coup (in Mali) and there is an organized and methodical anti-French discourse going on," Colonna said. "Potentially, yes, one can imagine a link in this anti-French campaign and Wagner."
French diplomats have said that Burkina has forged relations with Wagner while neighbouring Ghana's president alleged in December that Ouagadougou had hired the group. Burkinabe authorities have not responded to Reuters' requests for comment.
In its response, Wagner has accused Western governments, their intelligence services including the CIA and armed U.N. peacekeeping missions of committing some of the offences Wagner has been accused of in Africa, including amassing mineral resources through illegal mining and orchestrating coups.