Putin and Battulga are in Qingdao for the 18th Shanghai Cooperation Organization summit.
May - whose government has accused Russian Federation of being behind the poisoning of a former spy in Britain earlier this year - said that the G7 had to be convinced that Moscow had changed.
The medal is the latest in a series of public gestures of admiration between the pair.
He said he did not discuss a possible meeting venue with Trump but "many" countries including Austria were keen to help in organising such a summit.
As the Global Times recalled, Putin presented Xi with the Russian Order of St. Andrew previous year, one of the highest honors Russia can bestow upon a foreign leader.
Putin's remarks, however, were notably less effusive.
Xi then awarded Putin China's first ever friendship medal, on a large gold coloured chain, an event carried live on state television. "That is not good for us nor for any of our friends in the world", Macron tweeted.
"We just had some vodka together and carved some sausage", he recalled, adding: "I have never had such a relationship or arrangement with any other foreign colleague".
Thanks to President Putin, bilateral ties have withstood the test of ever-changing worldwide situation, Xi said, noting the deepened political and strategic mutual trust, great progress in pragmatic cooperation in various fields, and the enhanced friendship between the two peoples.
It may also add to concerns in the United States and among allies that the U.S. President, whose campaign team is under investigation for possible collusion with Russian Federation, is too close to the Kremlin.
At a press conference held subsequent to the signing ceremony, President Xi praised the ever expanding levels of bilateral trade between China and Russian Federation which this year hit an all time high.
"Canada has already raised cases at the WTO and at NAFTA and we will retaliate but we say that with great sadness especially here", she said.
Last month, Trump pulled out of former President Barack Obama's landmark 2015 nuclear accord with Iran over the objections of European allies and other nations. Tension between President Donald Trump and traditionally close US allies has overshadowed that global meeting.
Russian Federation supports China's proposal for a "dual track" and "suspension-for-suspension" approach that calls for North Korea to stop its nuclear and missile testing in exchange for a halt to joint military exercises between the USA and South Korea.
Russian Federation joined the group in the 1990s after emerging from the wreckage of the Soviet Union, making it the G-8, but its armed intervention in its neighbor Ukraine in 2014 and seizure of the Crimean peninsula angered other major powers.