MOSCOW (BLOOMBERG) - After US sanctions crippled an entire Russian industry and air strikes in Syria threatened the first direct clash between nuclear superpowers since the Cold War, President Vladimir Putin is seeking to dial down the tension.

Russia's leader wants to give President Donald Trump another chance to make good on pledges to improve ties and avoid escalation, according to four people familiar with the matter.

One said the Kremlin has ordered officials to curb their anti-US rhetoric.

Mr Putin's decision explains why lawmakers on Monday (April 16) suddenly pulled a draft law that would've imposed sweeping counter-sanctions on US companies, two of the people said.

The relatively limited nature of the weekend attacks on alleged chemical weapons facilities in Syria, where Russia is backing government forces in a civil war, was seen as a positive sign in the Kremlin, considering Mr Trump's ominous tweets announcing missiles would soon be flying.

"Putin is ready to make numerous, deep concessions, but he has to appear like he's not losing," said Mr Igor Bunin of the Centre for Political Technologies, a consultancy whose clients include Kremlin staff. "He understands Russia can't compete with the West economically and he doesn't plan to go to war with the West."

The Kremlin is still coming to grips with the economic impact of the most punitive penalties the US has imposed since first sanctioning Russia four years ago, over the conflict in Ukraine.

The latest measures, which Treasury called payback for Mr Putin's "malign activity" in general, hit one of the country's most powerful businessmen, billionaire Oleg Deripaska, the hardest.

Shares of Mr Deripaska's aluminium giant Rusal have plunged about 70 per cent in Hong Kong since the US basically banned the company from the dollar economy on April 6, erasing about US$6 billion (S$7.87 billion) of value and threatening 100,000 jobs at a time when Russia is limping out of its longest recession in two decades.

Mr Putin, who is due to be sworn in for what may be a final six-year term next month, is keen to avoid having another major company suffer a similar fate.

It could be too late to reverse the downward spiral that's taken relations to the lowest level in decades.

While Mr Trump is open to trying to improve ties, Congress and much of his administration are committed to keeping the pressure up on a country many view as America's No. 1 enemy after allegations of Kremlin meddling in the 2016 elections.

Further complicating the diplomatic dance are the often-conflicting signals coming from the White House.

Mr Trump's reported decision on Monday to put the brakes on new Russian sanctions triggered a brief rally in the rouble, which fell the most since June 2015 last week.

On Tuesday, for example, economic adviser Larry Kudlow sent Russian markets reeling when he said "additional sanctions are under consideration".

Still, the Kremlin is holding back from further escalation.

On Monday, legislators abandoned - at least for now - a Bill that would have limited a broad range of trade with the US, from farm products and medications to aviation and space.

One of the most outspoken critics of the Pentagon, Deputy Prime Minister Dmitry Rogozin, said a proposed ban on selling rocket engines "would hurt Russia more than the US", given Moscow's dependence on American contracts.

Even before the US attacks in Syria, the Kremlin told officials to tone down threats of retaliation.

Russia's ambassador to Lebanon doesn't seem to have gotten the message. In an interview with local television that made headlines the world over, he said Russia wouldn't hesitate to shoot down missiles and attack whatever platforms they were launched from if its forces were threatened.

The Kremlin was livid, according to one official, and the warning wasn't repeated.

With the initial euphoria over Mr Trump's surprise election victory long over, officials say they're convinced there's little the US President can do to overcome what they see as entrenched anti-Russian bias among a Washington establishment that's determined to maintain global dominance at any cost.

Last month's waves of diplomatic expulsions throughout the West over the nerve agent attack on a Russian turncoat spy in the UK accelerated a breakdown in relations that's been building for years.

Still, officials are holding out hope that Mr Trump might be able to stop the downward slide, especially after he congratulated Mr Putin on his re-election in March and dangled the prospect of a White House summit.

To be sure, plenty of areas of tension remain, including over alleged Russian hacking, trolling and other forms of online aggression.

US and UK officials this week issued a rare joint warning of what they called stepped up Russian probing of corporate and government computer systems in the West.

"Russia is our most capable hostile adversary in cyber space," said Mr Ciaran Martin, chief executive officer of Britain's National Cyber Security Centre.

The alert included advice to companies about how to protect themselves and warned specifically of attacks on routers, the devices that channel data around a network.

While reluctant to give precise details of the threat, officials said once a router is hacked, it can be used to capture data and even attack other computers, potentially overwhelming the Internet.

But Mr Vladimir Frolov, a former Russian diplomat who's now a foreign policy analyst in Moscow, said such cyber activities don't usually weigh on state-level negotiations since all countries are pursuing the same capabilities.

"Cyber espionage is considered a legitimate activity, it shouldn't really affect bilateral relations," he said.