In a statement today at the Shanghai Cooperation Organization, Putin said it wasn't Russian Federation who left the industrial coalition.
Reports said the Kremlin has reached out to Austria about hosting a meeting between Trump and Putin sometime this summer.
Putin said that he believes Trump is a "thoughtful person" who can listen and respond to meaningful arguments.
Appearing on Rossiya-1's Saturday News with Sergei Brilev show on Saturday, Putin said that Trump, during his election campaign, had promised to improve ties with Moscow.
Trump also added he has been Russia's "worst nightmare".
"The president of the United States himself has repeatedly said that he considered such a meeting appropriate, and I confirm that this is indeed the case. Please, we will be glad to see everyone here in Moscow", Putin told reporters at a briefing in China's city of Qingdao.
"We should reject self-centered, shortsighted and closed-door policies", said Xi, whose own country has been accused of restricting broad access for foreign firms to its huge market.
Adding an extra layer of friction to the gathering, Trump suggested that Russian Federation - which was expelled following the annexation of Crimea - should be allowed to rejoin the group.
Trump riled USA allies on Friday when he suggested Russian Federation be readmitted to the G7 organization, despite the country's 2014 expulsion because it annexed Crimea from Ukraine.
"As for Russia's return to the G7, G8 - we never left it", he said.
Before the medal ceremony, Putin said the two heads of state had enjoyed "fruitful" talks. "It's time to stop that chatter and deal with real issues".
Mr Trump announced back in March that he and Mr Putin planned to meet soon but since then relations between Washington and Moscow have since deteriorated over the conflict in Syria and the Salisbury spy poisoning scandal.
Professor Zhu Feng, director of Nanjing University's Institute of International Studies, also said the award was a clear signal to the U.S. that if it increased pressure on both countries, it would mean both China and Russian Federation would have no choice but to increase cooperation so they can jointly deal with Washington.
It is disappointing that the United States unilaterally withdrew from the Iran nuclear deal, said the statement, adding that China and Russian Federation will try their best to safeguard the deal and take note of the extreme importance of protecting the interests of all parties who engage in economic and trade cooperation with Iran from unilateral imposition of long-arm jurisdiction.
Russia's President Vladimir Putin and Iranian President Hassan Rouhani met Saturday to discuss regional issues including both countries' involvement in the Syrian civil war and the USA withdrawal from a landmark nuclear deal with Iran.
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