Telegraph - Mr Putin gathered top advisers in Moscow the day before a telephone call with the new US president in which they are expected to touch on a number of issues including sanctions relief and fighting terrorism.
Speaking at a press conference with Theresa May on Friday night, Mr Trump said it was "too early" to be talking about whether sanctions would be lifted, but that a good relationship with Russia would be a "positive not a negative."
"I don't know the gentleman. I hope we have a good relationship but it is very possible we won't," he said of Mr Putin. Earlier Kellyanne Conway, a senior advisor to Mr Trump, told Fox News that cooperation in fighting terrorism and sanctions Barack Obama imposed on Russia following the annexation of Crimea and invasion of east Ukraine in 2014 would be discussed.
"All of this us under consideration,” she said when asked whether sanctions would be on the table during the call.
"Certainly in addition to improving relations with different foreign leaders and their nations around the globe, if Vladimir Putin wants to join with the US to have a serious conversation about how to defeat radical Islamic Terrorism,” she added.
Ms Conway’s comment came amid unconfirmed reports that Trump staffers have already drafted an executive order that would lift some of the sanctions imposed by Mr Obama.
Any move to lift sanctions would set Mr Trump on a collision course with the Republican-controlled Congress and with key allies in Europe, including the UK and Germany, which back maintaining the EU sanctions regime.
John McCain, the senior republican senator, said Mr Trump should reject what he called a "reckless course.”
"If he does not, I will work with my colleagues to codify sanctions against Russia into law," he said.
Sanctions passed into law would require an act of congress to cancel, limiting Mr Trump’s ability to reverse them.
At the press conference with Mr Trump, Mrs May reiterated that Britain believed sanctions against Russia should remain in place until the Minsk peace agreements are implemented.
Angela Merkel, the German Chancellor, and Francois Hollande, the president of France, are also expected to speak to Mr Trump on Saturday.
A source told Reuters that Mrs Merkel’s call will focus on Russia. Mrs Merkel is set to host Petro Poroshenko, the Ukrainian president, in Berlin on Monday.
The Kremlin sought to play down expectations of a sudden breakthrough in relations on Friday.
"You can hardly expect a substantive discussion on all issues from the first telephone call after President Trump’s inauguration,” said Dmitry Peskov, Mr Putin’s spokesman, on Friday.
Asked about the reports of a plans to drop sanctions, he said: "I do not know how far these statements correspond to reality.”
Saturday’s call will be the two leaders’ first conversation since Mr Trump was inaugurated. They spoke briefly when Mr Putin congratulated Mr Trump on his election victory in November.