Regulations state a team must replace an injured rider after two races, but Brivio has insisted there is no rush, saying: "We will miss the next couple of races because that is allowed.
"If it were up to Yamaha to decide alone, we would not enlist anyone to take his place from now to the end of the championship."
Spaniard Jorge Lorenzo will now be the team's solitary rider at the UK race at Silverstone next week and at Assen in the Netherlands on 26 June.
Valentino Rossi breaks leg in horror crash
Brivio added: "If, as I think, Valentino stays out for longer [than two races] we have to find a replacement.
"It's very strange to speak about Valentino's replacement. But we will try to find the best solution we can."
The decision to delay appointing a new rider is likely to have been influenced by a lack of obvious replacements for the phenomenally successful Italian, who had not previously missed a race in 15 years of racing.
Brivio has already stated that the team's test riders are unsuitable while American superbike world champion Ben Spies is under contract to the non-works Tech3 Yamaha MotoGP team.
Despite the severity of the injury, Rossi has already quashed any rumours that he might not make a return to racing.
"Do not worry, I'll come back soon. I was afraid, now it's a lot better," he said on Saturday.
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