The nature of his illness has not been disclosed, but sources have stressed it is “nothing to worry about”.
Hours earlier, Clooney had been seen arriving at the Lido with Dern, dressed in a navy suit, striped shirt and tinted sunglasses.
His health scare comes after he recently discussed his acting plans.
In March, he told CBS News programme 60 Minutes that he would no longer appear in romantic comedies.
“Look, I’m 63 years old. I’m not trying to compete with 25-year-old leading men. That’s not my job. I’m not doing romantic films anymore.”
He was speaking before his Broadway debut in Good Night, And Good Luck, an adaptation of his 2005 Oscar-nominated film of the same name.
He plays journalist Edward R. Murrow in the stage production, which recounts Murrow’s televised confrontation with Senator Joseph McCarthy during the “Second Red Scare” of the 1950s.
He said: “It’s exciting to be [on Broadway.] Let’s not kid ourselves. It’s nerve-racking. And there’s a million reasons why it’s dumb to do.
“It’s dumb to do because you’re coming out and saying, ‘Well, let’s try to get an audience to take this ride with you back to 1954’.”
Reflecting on the relationship between the Government and the press, he said: “We’re seeing this idea of using Government to scare or fine, or use corporations to make journalists smaller.
“Governments don’t like the freedom of the press, they never have. And that goes for whether you are a conservative or a liberal or whatever side you’re on. They don’t like the press.”
The lifelong Democrat also spoke about his decision to call on Joe Biden, 82, to step aside from the last US presidential race after his disastrous debate with Donald Trump, 79.
“I was raised to tell the truth,” he said. “I had seen the President up close for this fundraiser, and I was surprised. And so, I feel as if there was a lot of … cowardice in my party, through all of that, and I was not proud of that, and I also believed I had to tell the truth.”