The United States has to make it clear to Russian President Vladimir Putin that there are lines that the U.S. will not allow Russia to cross, former Defense Secretary Leon Panetta told CNBC on Tuesday.

"There's no question we're in a new chapter of the Cold War with Russia," Panetta said on "Closing Bell."

"There are an awful lot of steps that the Russians are taking to be very aggressive," he said. "Part of it is because they've read weakness into the United States, as well as our allies."

Tensions have been escalating between the West and Russia over the civil war in Syria. Last Friday, a U.S.-led airstrike targeted Syria's chemical weapons infrastructure. It came one week after an alleged chemical weapons attack by Syrian President Bashar Assad that killed at least 60 people.

Putin, who supports Assad's government, warned Sunday that further Western attacks on Syria would bring chaos to world affairs.

Russia's involvement in Syria is just the country's latest act of aggression, suggested Panetta, who also served as CIA director under President Barack Obama and as chief of staff to President Bill Clinton.

Russia came under fire for annexing Crimea in southern Ukraine and is accused of meddling in the 2016 U.S. presidential election. In a March interview with NBC News, Putin denied U.S. charges that he was behind the interference in the election which resulted in Donald Trump becoming president.

Panetta believes the U.S. needs to show its power, which could open up an opportunity to have discussions with Putin.

"The only way you deal with Putin is from strength, not from weakness," he said.

On Sunday, Nikki Haley, U.S. ambassador to the United Nations, said new sanctions were coming to punish Russia for its support of Assad's government. However, on Monday, Trump delayed further action, Reuters reported.

Trump's top economic advisor, Larry Kudlow, denied there was confusion over the issue.

"I think the issue here is we have a set of sanctions and additional sanctions are under consideration but have not been determined," Kudlow, a former CNBC senior contributor, told reporters on Tuesday.

— Reuters contributed to this report.