But two months into his presidency, reality has so far proved otherwise -- with the Trump administration talking tough and taking diplomatic positions akin to those of his predecessor Barack Obama, experts say.
The "bromance" between Putin and Trump was abruptly adjourned amidst a political storm that continues to rage in Washington over alleged collusion between Trump's inner circle and the Kremlin during the 2016 election campaign.
Ongoing Russian operations in Ukraine have also forced Trump, who famously touted Putin as being a "stronger leader" than Obama, to moderate his stance.
'Ongoing hostility' in Ukraine
US Secretary of State Rex Tillerson and Defence Secretary James Mattis made the White House's position on Moscow clear with during trips to Europe last week.
Tillerson, who as Exxon chief had maintained close business ties to Moscow, on Friday reaffirmed Washington's support for Ukraine.
Kiev is battling pro-Moscow rebels in its east and lost Crimea three years ago when Russia annexed it.
Meeting with fellow NATO foreign ministers for the first time, Tillerson employed deliberately strong words, denouncing Moscow's "ongoing hostility and occupation" in Ukraine.
He said Washington would stick with sanctions against Russia laid out under Obama's secretary of state, John Kerry, until "Moscow reverses (its) actions" and cedes Crimea.
"American and NATO support for Ukraine remains steadfast", he said, using a diplomatic line often repeated by Kerry.
In a closed-door NATO luncheon, Tillerson also drew standing applause when he told attendees that Russia could was "no longer trusted" and had to decide "if it wants to engage in the world or be isolated", according to a US State Department official.
For Jeffrey Rathke at the Center for Strategic and International Studies, a former press secretary in Kerry's State Department, the Trump administration is hewing more closely now to the Obama playbook on Russia.
"I think Tillerson's (and Mattis's) strong lines do reflect a pretty high degree of continuity," he told AFP.
He said the current controversy over Russian interference in the election and ties to the Trump camp made it "extremely difficult" for the White House to change course.
"For now we have inertia that keeps the outline of the Obama policy in place," he added. "But no one is sure how long that will last."
'Mucking around' in elections
Pentagon chief Mattis also let loose on Moscow during a visit to London on Friday.
He said "Russia's violations of international law are now a matter of record", citing both the Crimea annexation and "other aspects of their behaviour in mucking around inside other people's elections" -- without specifically mentioning the US vote.
"The point about Russia is they have to live by international law just like we expect all nations on this planet to do," Mattis said.
He also rebuffed recent calls by Moscow to step up intelligence and military cooperation with the Trump White House, saying "we are not in a position right now" to do so.
The Kremlin, in response, criticised NATO "slander" against it and said it was "perplexed" by Tillerson's remarks.
Its confusion is not surprising. Trump not long ago sounded a very different note.
On the campaign trail he said NATO was possibly "obsolete", and called for better cooperation with Russia, notably in the fight against the Islamic State group in Syria.
On a personal level, the reality TV star and real estate tycoon also lauded Putin as "very smart", as someone he would "get along very well" with and -- in a 2013 tweet -- wondered, "Will he become my new best friend?"
But now, in a February 5 interview with Fox News, the president was less effusive, saying of Putin that he "respect(s) him... but that doesn't mean I'm going to get along with them".
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