Red Bull motorsport consultant Helmut Marko, a key figure at the two teams owned by the Austrian energy drink company, told Germany's Auto Bild the decision had been taken.
"Kvyat will not return again. We do not believe he can make the turnaround in the long term," said the Austrian ahead of this weekend's Mexican Grand Prix.
"Gasly and Hartley are finishing the season. Then we'll see what happens."
Red Bull-owned Toro Rosso, who are switching to Honda engines next year, have yet to confirm their 2018 lineup but have limited options from within their own stable.
The team acts as a feeder for Red Bull Racing and has served as a springboard for four-times world champion Sebastian Vettel and race winners Daniel Ricciardo and Max Verstappen.
At 27-years-old, and with a Le Mans 24 Hours title and world endurance championship under his belt, New Zealander Hartley hardly fits the mould of a young driver but he has seized his chance.
Dropped by Red Bull after testing with Toro Rosso as a teenager in 2009, the Kiwi returned to the fold and made a strong debut in Texas last weekend alongside Kvyat, 23.
Gasly, last year's GP2 champion, made his debut in Malaysia this month but missed the U.S. Grand Prix to try and win the Japanese Super Formula series that he had previously been competing in.
He returns for Mexico, with Hartley retained.
Kvyat has now been dropped three times by Red Bull.
Promoted from Toro Rosso to Red Bull at the end of 2014, he was demoted after four races of last season to make way for Verstappen.
This year he sat out Malaysia and Japan for Gasly and returned for Austin when Spaniard Carlos Sainz, Toro Rosso's main points scorer, switched to Renault as part of an engine deal.
Kvyat's only real option in Formula One now appears to be Williams, who could have a vacancy if they decide not to retain Brazilian veteran Felipe Massa.
"Formula One is not everything. There are other good opportunities in this life," the Russian told Sky Sports television last weekend as speculation swirled around his future.
"I'm always going to be a racing driver and I always want to race. But there are always good places to be somewhere."
(Reporting by Alan Baldwin, editing by Christian Radnedge)
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