Earlier, the Russian said he would be patient while the Red Bull junior team sorted out the identity of its engine supplier for next year.
So with the Honda deal now done, Kvyat said in Singapore: “I’m waiting for them to have the time to think about me.
“Today is not the time as the main thing is the racing this weekend,” he added, “but soon we will have time to clarify the situation and get answers.”
Kvyat, 23, said he thinks Toro Rosso’s switch from Renault to Honda next year is “good for the team”.
“Our team has constantly changed engines in the last three years,” he admitted, “but it’s great that we now have clear plans for the future and will be able to build strong relationships with a supplier.
“We will be the only team with Honda engines, so it’s a fantastic opportunity to build a solid foundation.”
Kvyat’s future is actually far from clear, even though Toro Rosso might want continuity given Carlos Sainz’s departure to Renault.
However, Pierre Gasly is expected to replace Sainz next year, and in Singapore Sean Gelael – the son of the Indonesian KFC magnate – drove in free practice.
Not just that, Toro Rosso boss Franz Tost sounds open to accommodating a Honda-backed driver next year, such as Nobuharu Matsushita.
“I assume Daniil Kvyat and Carlos Sainz will finish the season with us,” said Tost in Singapore.
“Regarding the drivers for next year, Red Bull will decide and of course we will also take into consideration any request from our partners.”
When asked about that, Kvyat said: “I do not want to comment on this.
“In formula one, situations can change very, very quickly, so it’s too early to start worrying or fearing. I think that is the right approach for me.”