(CNN) A Donald Trump supporter and former member of Congress is in Russia this week, as questions about the President-elect's policies toward Moscow roil his transition.

Former Georgia Rep. Jack Kingston spoke this week in the Russian capital with American business leaders in a closed-door session. Publicly, however, he suggested that Western sanctions on Russia over its aggression in Ukraine and annexation of Crimea could be lifted.

"Trump can look at sanctions. They've been in place long enough," Kingston said during an interview with NPR in Moscow. "Has the desired result been reached? He doesn't have to abide by the Obama foreign policy. That gives him a fresh start."

A spokeswoman for Trump did not immediately respond to a request for comment on Kingston's visit or remarks.

The sanctions resulted from a major push from US to coordinate with its European allies, who might see the move as an abandonment of cooperation with Europe. At the same time, the Europeans have been less enthusiastic about the economic restrictions because of their trade ties to Russia and would be unlikely to extend them if Trump no longer backs them.

"Among the many questions the businesses asked ... they had a number of questions, but one of the questions was 'What about sanctions?' to which I said 'Sanctions not something the administrations going to lead with at all,'" Kingston told CNN's Jake Tapper on "The Lead" Thursday. "The sanctions have been in place a while now. The administration should take a look and say, 'Are the results what we were looking for?' Brand new administration, it's a good time to revitalize."

Back in the US, Kingston's former colleagues in Congress are wrestling over how to respond to Russia's alleged interference in the presidential election -- with many, Republicans included, calling for additional sanctions due to US intelligence assessments of hacking of Democratic organizations.

One of the consistent messages of Trump's presidential campaign was a promised push for better relations with Russia, in particular its President Vladimir Putin.

In nominating ExxonMobil CEO Rex Tillerson to lead the State Department, Trump again signaled his desire for a diplomatic thaw through the pursuit of shared economic and foreign policy goals. Tillerson, who did extensive business in Russia and was recommended to the transition by a group of Republican establishment foreign policy hands who do business with the oil giant, was awarded the Order of Friendship by Putin in 2013.

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But Tillerson's nomination has set off alarm bells among some Senate Republicans. South Carolina Sen. Lindsey Graham told CNN's Wolf Blitzer on "The Situation Room" Wednesday, "I can't imagine I would vote (to confirm) anybody that believes that we should not sanction Russia, given the fact that they did in fact interfere in our election."

"If they don't believe sanctions are appropriate, given what Putin has been doing all over the world, including in our backyard, then I don't think they have the judgment to be secretary of state," he added, "because if you don't go after Russia, you're inviting the other bad actors on the planet to come after you."

On Thursday, Graham's Democratic colleague, Missouri Sen. Claire McCaskill, tweeted that she was "down" with his tweeted pledge to put "crippling sanctions against Russia" on Trump's White House desk.

Controversial Trump loyalist Carter Page, who has spoken positively about the Tillerson nomination and opposes sanctions, also raised eyebrows recently when he turned up in Moscow and gave a presentation and news conference at the Russian state news outlet Sputnik.

"If you look at where the current state of US-Russia relations is, it's much more toxic and much more dangerous and there's a lot more work that needs to be done to overcome this," Page said, before suggesting reports on US intelligence assessments about Russian hacking during the campaign "almost borders on the 'fake news' category."

Asked by a reporter if he believed there was a deliberate US government attempt to deceive the American public about the source of the hacks, Page said it is "very easy to make it look like it was country X, in this case, Russia, that did this."

"I've talked with various IT experts that have suggested that (intentional US misdirection) could very well be a serious possibility," he added, "and these guys are pros to make certain paths that can mislead and again we've seen many mistakes from an intelligence standpoint previously."

On Thursday, Trump still publicly questioned whether Russia was involved in hacking American politicians, an assertion uniformly supported by the US Intelligence community.

"If Russia, or some other entity, was hacking, why did the White House wait so long to act? Why did they only complain after Hillary lost?" Trump tweeted