In an interview with Sunday's edition of the Oesterreich newspaper, Strache said that "a rethink by the European Union would be desirable, because the sanctions have above all damaged our Austrian economy".
Putin said: "We spent a lot of money and I agree it needs to work for the benefit of sport".
Putin adds "there will certainly be other contenders, but only the best will win".
Putin, making his first foreign trip since being sworn in for a fourth term, said the restrictions are "harmful for everyone - those who initiated them and those who are targeted by them".
Over the weekend, Strache - whose party in 2016 signed a cooperation agreement with the main Kremlin party, United Russia - said it was time to lift sanctions against Russia. USA and Russian officials haven't discussed specifics yet.
As for the exploration of more distant space, Putin said that Russian Federation plans to test a new super-heavy rocket in an unmanned mode in 2022 and launch it with astronauts on board in 2024. "They see that Russian Federation has become a competitor to them ... we propose that this is a very mistaken policy". Putin asked during his annual phone-in show, referring to Canada and Mexico.
The poisoning of former Russian double agent Sergei Skripal and his daughter in Britain further strained the EU-Russia ties.
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Former champion Novak Djokovic sprinted past Spain's Fernando Verdasco 6-3 6-4 6-2 to reach his 12th Roland Garros quarter-final. Cecchinato's ranking of No. 72 is the lowest in a decade for a man in the final eight at the French Open .
Amazon to show 20 live Premier League games
Pay-TV giant Sky has for years been the home of Premier League games in Britain, which are a cornerstone of its business. This is a big deal though, because someone managed to break through Sky and BT's hold on Premier League matches.
Aide who dismissed McCain could resurface in administration
The effort to find leakers has led to an acrimonious atmosphere at the White House, aides have told CNN . White House aide Kelly Sadler is out of a job nearly a month after she reportedly mocked a "dying" Sen.
Asked about the poisoning during his annual call-in show on Thursday, Putin said he does not want to comment on the cause of their illness and said Russian Federation is still seeking consular access to both of them and to their case files.
Mr Putin also said the government will be looking to streamline the tax system in order to fight poverty, but denied speculation that Russian Federation was considering abandoning its flat-rate income tax. "It was invented to limit the lawful rights of the people who live in those territories", the president said.
Responding to speculation that authorities might be looking to scrap the flat income tax, Putin said his government has decided that this is "not expedient" at this point.
Austria says it wants to act as a bridge between Russian Federation and the EU.
On Thursday he said Russia's gross domestic product is now 1.5% higher than a year ago.
But he said that Pyongyang will demand "absolute security guarantees" if the United States sticks with its demand that Kim must totally abandon nuclear weapons to obtain relief from sanctions.