Spanish media have reported Hazard as a transfer target for Real Madrid throughout the close season.
When asked about the prospect of a foreign team making a move for Chelsea players, Sarri said: "It's impossible because, here, the market is closed. We cannot change important players now.
"Hazard is with us. I think, definitely. Not a problem. I have spoken with him three or four times for everything. He never said something about the market, so I think he's very happy to stay here."
Arrizabalaga has the daunting task of replacing World Cup Golden Glove winner Thibaut Courtois, who established himself as one of the finest goalkeepers in the Premier League after arriving at Stamford Bridge in 2011.
Sarri conceded Arrizabalaga will need time to adapt to the English game but praised Chelsea for acquiring "one of the most important" young goalkeepers in Europe.
"I'm very happy for both because Courtois' situation was clear," Sarri said.
"He wanted to go to Madrid so we lost maybe one of the most important goalkeepers in the world. But we bought a young goalkeeper, one of the most important among the young goalkeepers in Europe, so I'm really happy.
"(Also) for Kovacic, because he has the right characteristics for us. He is a very technical player who can play between the lines of the opponent. I'm very happy and now I think the squad is complete."
Chelsea will kick off their league campaign against Huddersfield Town at the John Smith's Stadium on Saturday.
(Reporting by Hardik Vyas in Bengaluru; Editing by Christian Radnedge)
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