Pascal Vasselon, technical boss of the recently-renamed Toyota Gazoo Racing Europe operation, made the admission during an Instagram live session organised by the organiser of the Le Mans 24 Hours, the Automobile Club de l'Ouest.
Toyota had been aiming for a summer debut for its hypercar before the global coronavirus pandemic disrupted its preparations, with the delay to the start of the 2020/21 WEC campaign until March 2021 giving the Japanese manufacturer an extra six months of preparation time.
"The initial idea was to start in July with the first [tests], but today's situation has not allowed us to do that," said Vasselon.
"The fact that the first race of 2021 will be in March still gives us some time to work well; the hypercar will be on track in October."
A Toyota spokesperson confirmed to Motorsport.com that the design work on the marque's GR Super Sport-based contender has been completed, but manufacturing has been delayed owing to a shortage of some components from particular suppliers.
Carbon fibre in particular has been hard to source as Italy, one of the European countries worst affected by COVID-19, is one of the major suppliers of the material.
Design and engineering work at Toyota's Cologne base has been able to continue more or less normally with many staff working remotely, while a shift system has been implemented for production when the required materials are available.
The COVID-19 crisis has impacted both the current WEC season and the following one, which had originally been due to begin at Silverstone in early September.
Vasselon admitted late last year that Toyota's timeline for having the car ready for the start of the 2020/21 season was "risky", not helped by the hypercar rules only being finally firmed up in the weeks leading up to last year's Le Mans 24 Hours.