Washington (CNN) George W. Bush tried it. Barack Obama and Hillary Clinton tried it. Now Donald Trump is vowing to reset relations with Russia.

But could the unintended costs outweigh the benefits?

The President-elect's Russian gambit seems like a solid foreign policy goal, given that the estrangement between the rivals -- owners of the world's biggest nuclear arsenals -- is deeper than at any time since the Cold War.

An improvement in relations could yield gains for the United States in reviving non-proliferation projects and closer anti-terrorism cooperation.

But there are big questions about whether Trump will be any more successful than his predecessors, whose initial inroads eventually foundered largely due to the behavior of President Vladimir Putin, and whether he will end up emboldening a leader that many -- including key Republicans in Congress -- see as a top US adversary.

Trump has made no secret of his admiration for the Russian strongman, once saying he was a better leader than Obama, and on the campaign trail showed little concern about resurgent Russian influence. By nominating Rex Tillerson as secretary of state , Trump has pleased Moscow, where the ExxonMobil CEO has good relationships with senior officials including Putin.

And even some Russia skeptics can see the merit in trying to improve relations.

"This could be the time for us to make a move and work things out with Russia to some extent," New York Rep. Peter King told CNN's Wolf Blitzer on Tuesday. "I am confident that Donald Trump will not be taken in -- there is some room for negotiations with Russia here, but we should do it with a strong hand."

Will Trump turn a blind eye?

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But Trump's determination to find new areas where the two sides can work together could necessitate that Washington turn a blind eye to behavior by Putin on issues like human rights and the central rationale of his foreign policy -- restoring Russian influence at the direct expense of the United States.

Indeed, easing the suspicion between Moscow and Washington could come with some serious negatives.

An improved relationship with Russia might require the United States to drop its opposition to the Kremlin's annexation of Crimea, and the lifting of sanctions against top Russian officials to punish the land grab from Ukraine.

In Syria, the US may need to abandon its support for moderate rebels and effectively align itself with a regime accused of war crimes and behind brutal violence in Aleppo. NATO members may be further shaken by a rapprochement between the White House and the Kremlin following Trump's campaign trail critiques of the alliance.

If moves like lifting sanctions occur, the Trump administration would establish a precedent that could embolden Russia's attempt to throw its weight around in the rest of the world.

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Steven Pifer, a Brookings Institution scholar and former US ambassador to Ukraine, said the effects of Russian aggression in the Eastern European country go far beyond that conflict.

"Simply accepting Russian behavior that is truly egregious" would prove problematic to US efforts to halting Russian meddling around the world, according to Pifer. "If we were to recognize Crimea as Russian, that would put us in the likes of a handful of countries like North Korea, Syria and Venezuela," he said.

"Are we prepared to let the Russians get away with the actions of eastern Ukraine?" he said, adding that would concede a sphere of influence to the Russians and rattle the nerves of US allies in the Baltic region.

To revive relations with Moscow, President Trump would also have to shrug his shoulders at any findings from congressional and White House-mandated investigations into allegations that Russia hacked the presidential elections. His refusal to believe assessments by US intelligence agencies that Moscow intervened in the presidential election, though, suggests that would not be a problem for the new occupant of the Oval Office.

Could Trump lift Russia sanctions?

Trump would not even have to consult with US allies over lifting sanctions. He could accomplish that by a pen stroke since they were mandated by Obama in a series of executive orders.

The choice of Tillerson, who is on the record as opposing sanctions -- which have cost ExxonMobil lucrative energy exploration contracts in Russia -- is being read as another hint by Trump of a sharp shift in US policy toward Moscow and has not gone down well at the current White House.

"Throughout his campaign, the President-elect indicated his intent, if elected president, to pursue warmer relations with Russia. So what better way to do that than to choose somebody who's been awarded the Order of Friendship by Vladimir Putin to be your secretary of state?" spokesman Josh Earnest said Tuesday.

It's possible that the geopolitical and strategic considerations that shape the US-Russia relationship will dictate the pace and breadth of Trump's engagement, regardless of his past statements or future intentions.

"I think US policy especially with respect to a big geopolitical actor like a Russia or China is more often driven by the shape of the broader world and US and Russian interests in it than it is driven by the individual proclivities of a US president or a Russian president," said Matthew Rojansky, a Russia expert at the Wilson Center.

Syria: An opportunity and a risk

The central focus of Trump's work to improve the US relationship with Russia could come in the Middle East, where Trump has said the US needs to urgently reevaluate its strategy in fighting ISIS, particularly in Syria.

"Wouldn't it be nice if we got along with Russia?" Trump said repeatedly on the campaign trail. "Wouldn't it be nice if we got together with Russia and knocked the hell out of ISIS?"

The US has repeatedly sought to forge agreements to both join forces with Russia in combating ISIS and to bring an end to the Syrian civil war, efforts that last broke down in October over Russia's ongoing efforts to prop up the Syrian regime and its attacks on US-backed moderate rebels.

In his first post-election interview with The Wall Street Journal, Trump criticized the US's backing of moderate rebels fighting the Syrian regime and suggested he was more interested in fighting ISIS alongside Russia than pushing for the ouster of Syrian President Bashar al-Assad.

Trump has sought to simplify the US's role in the conflict in Syria, boiling down the campaign to destroying ISIS and making everything else secondary to that objective.

Noting that the Russians have "one very simple objective, which is to help Assad," CATO institute senior fellow Doug Bandow argued the US could benefit from paring down its objectives in Syria.

"We want to get rid of Assad, stop ISIS, work with the Kurds, keep the Turks happy, only help the moderates," Bandow, who has argued against the US deepening its military involvement in Syria, said of the current situation.

He continued, "For all of Trump's lack of sophistication -- which is quite clear and is a problem -- he still has some basic understanding that priorities have to be set and you can't do everything."

But even cooperation on combating ISIS would raise serious problems for the US's standing in the Middle East and raise a slew of questions about how that would impact the decades-long battle for influence between the two countries in the Middle East.

The US has held the line on its calls for Assad to step down from power both because of the egregious human rights violations he has committed, but also with an eye toward greater US influence in the region. Ceding that point would ensure Russia can preserve a key ally in the region, strengthening its influence in the Middle East.

US support for Russia's role in Syria and in turn the Syrian government could also damage America's standing and popular support throughout the Middle East, particularly among allies who have decried the Syrian government's slaughter of tens of thousands of civilians.

Russians have gained influence in recent years through its support of the Iranian regime, which in turn has gained a sturdy foothold in Iraq in recent years.

Globally, a more accommodating US relationship with Russia could allow Moscow the space to rebuild the Cold War-era influence that the US spent decades deflating.