Trump's relationship with Russian President Vladimir Putin was once again in the headlines as the President drew sharp criticism from Republicans and Democrats alike by equating Putin's government with the US's.

The latest flap came when Fox News host Bill O'Reilly said "Putin's a killer" -- a reference to Putin's past as a KGB agent and instances in which his political opponents and critics have died suddenly -- and Trump shot back, "There are a lot of killers. We get a lot of killers. What, you think our country's so innocent?"

Trump's comments revived questions about his ties to Moscow, which is accused of interfering to help the Republican candidate win.

On Tuesday, Trump tweeted about the perceptions of his relationship with the Russian leader, writing in part, "I don't know Putin, have no deals in Russia, and the haters are going crazy..."

I don't know Putin, have no deals in Russia, and the haters are going crazy - yet Obama can make a deal with Iran, #1 in terror, no problem!

The outrage about Trump's many warm words for Putin has been fueled more by rhetoric than reality: So far, the administration has made very few policy announcements about how it will actually deal with Moscow.

And the few indications Trump officials have offered on Russia policy have either been contradictory or confusingly mixed:

-Administration officials have signaled radically different approaches to the sanctions in place against Russia for its aggressive support for separatist rebels in eastern Ukraine and its annexation of Crimea.

-They have both condemned Russia for its human rights violations and refused to do so.

Trump has questioned the need for NATO and voiced his strong support for the organization. And while Trump and his top security adviser have said they want better relations with Russia, they're also poised to support new NATO members despite opposition from Moscow.

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"There isn't anything yet resembling a formal policy from the administration," said Matthew Rojansky, director of the Kennan Institute at the Wilson Center. "The pieces don't yet add up to a particular deal in the offing."

Even so, lawmakers are preparing to get tougher on Russia and girding for a possible fight with the White House about that. There are ongoing Congressional investigations into ties between Trump, Russia and Moscow's interference in the election. And lawmakers from both parties support sanctioning Russia for its alleged hacking during the campaign.

Sen. Marco Rubio, a Florida Republican who sits on the Senate Foreign Relations Committee, is warning Trump and the White House that if they were to roll back sanctions on Russia without concessions from Moscow, the Senate would act. Rubio pointed to Russia's support for Ukrainian separatists that is destabilizing that country.

"There's a real threat of lifting sanctions minus the respect of Ukrainian sovereignty, and Putin meeting those conditions," Rubio said, speaking with CNN's Manu Raju about the sanctions legislation. "My sense is that we'd be able to pass that in the Senate and that we'd be able to pass that with a veto-proof majority."

The structure for creating a new US-Russia relationship isn't even fully in place yet. Trump and Putin had their first call on January 28, with the Kremlin and White House issuing general language about bilateral relations in need of repair.

But the White House has yet to put Russia-focused staff in place on the National Security Council or at the head of the relevant State Department bureau. Secretary of State Rex Tillerson, who has deep ties to Russia, was confirmed only last week. The State Department wouldn't comment on whether he has spoken yet to Russian Foreign Minister Sergei Lavrov.

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Once the new administration is fully up and running, it will have to engage on the many reasons for hostility between Russia and the old administration.

Beyond annexing Crimea, Moscow has worked to destabilize EU allies through fake news operations and hacking, lawmakers and former administration officials have said. Russian jets and submarines have harassed US allies in the Baltics, and, in November, Moscow announced plans to move nuclear-capable missiles closer to US allies in Eastern Europe.

Russian jets made several provocative and unsafe intercepts of US aircraft last year, and the US said Russia was responsible for the bombing of a UN aid convoy in Syria. Obama administration officials charged that Moscow either conducted the bombing or turned a blind eye as Syria did. US officials and lawmakers also charge that Russia has never really targeted ISIS in Syria, instead turning its guns on US-backed moderates.

Russia, for its part, has chafed at NATO's steady expansion toward its borders, a development it sees as a broken US promise. And Putin reportedly held the US in general and then-Secretary of State Hillary Clinton in particular responsible for rallies against him during 2012 Russian elections.

Despite the tensions, Trump took positions during the campaign that were staunchly pro-Russian, praising Putin and calling for better relations, saying, "Why not?"

But the early weeks of his administration have done nothing but muddy the waters on what his actual approach to Moscow will be.

After months of deriding NATO as "obsolete" and lacking a relevant mission, Trump has recently been executing what looks like a shift, after strong statements of support for NATO from Tillerson, Defense Secretary James Mattis and UN Ambassador Nikki Haley.

On Monday, Trump said he "strongly supported" NATO, while calling on member countries to boost their financial support for the alliance. Trump also has committed to attending a meeting of NATO leaders, to be held in Europe in May.

There's been a seeming shift on Ukraine as well. After suggesting on the campaign trail that he might recognize Russia's annexation of Crimea, the administration told the UN last week that it will keep sanctions against Russia in place until it leaves Crimea.

Haley's fiery speech to that effect came just a week after Trump adviser Kellyanne Conway had said the administration was considering easing sanctions against Russia.

Vice President Mike Pence repeated administration concern about eastern Ukraine on Sunday, telling ABC's "This Week" that the US is "very troubled" by escalating violence there and that Washington will be "watching" to see what Russia does.

Stephen Sestanovich, a senior fellow for Russian and Eurasian studies at the Council on Foreign Relations, said that there might be a silver lining to the Trump administration's mixed messages.

"I'm beginning to think that the administration's confusion and disarray could actually save it from serious mistakes," Sestanovich said. "Putin might like the idea of doing a deal with Trump, but who would do a deal with an administration that doesn't know what it thinks, how to act on it, or how to sell it at home?"