"We want to make sure not only that there is enough supply to go around, but more importantly that everybody has access to it across Europe," Gilead's Mercier said in a telephone interview.
The antiviral drug has been shown to shorten hospital recovery time in severe cases of Covid-19. Remdesivir and the steroid dexamethasone are the only drugs authorized in Europe to treat the novel virus.
Both have been given to US President Donald Trump, who is also receiving an experimental antibody cocktail.
Hospitalizations across Europe have been rapidly increasing, although in most countries still far below levels of the spring.
"Due to the increased hospital admissions, the demand for remdesivir is increasing rapidly," the Dutch spokesman said.
Read also: WHO: Europe to Brace for Uptick in Covid-19 Deaths in Coming Months
Poland's health minister Adam Niedzielski said on Tuesday the drug was in short supply in some hospitals. Its latest shipment from the European Union order arrived just last Friday.
Spain, which has Europe's highest infection rate, experienced shortages in late August, its medicine agency said. It has now enough doses to meet needs for the coming weeks, the health ministry said on Tuesday.
Britain, which has joined the European Union procurement for remdesivir, has rationed its supply, prioritizing Covid-19 patients who need it most, the health ministry said.
Shortages are likely to stir debate about the availability and pricing of Covid-19 drugs. While dexamethasone, a generic medicine, is widely available and cheap, remdesivir is protected by Gilead's patent.