The Reds have agreed to pay the Serie A club an initial £10m for the 26-year-old Italy forward, who has agreed a four-year contract.
Chiesa had been linked with Spanish giants Barcelona and a move to Saudi Arabia.
Liverpool registered their interest after it emerged that he was not part of Juve boss Thiago Motta's long-term plans.
Chiesa made 37 appearances, 29 of them as a starter, and scored 10 goals last season but his days have looked numbered since Motta's arrival.
Meanwhile, Chelsea forward Romelu Lukaku landed in Italy to finalise his transfer to Napoli. The fee for the Belgian striker is reported to be in the region of A$50m.
Brentford have signed teenage forward Gustavo Nunes from Brazilian Serie A side Gremio.
The 18-year-old has signed a six-year contract, with an option to extend the deal by two years, Brentford said.
"It is another good signing, we have fought with other clubs," Brentford coach Thomas Frank said.
"He is a winger who can go past players one-on-one. He is direct and he can create. Gustavo has a lot of potential and of course we need to maximise that in every aspect. He needs time to settle, but this is a very exciting signing."
Nunes joined Gremio's youth academy in 2021 and made his first-team debut in February this year.
With agencies
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