The imprisonment of Tymoshenko, who helped organise the June championships while in office and is the main political rival of Ukrainian President Viktor Yanukovich, has called into question Kiev's efforts to move closer to the European Union.
"I have left Ukraine's prime minister and president in no doubt that the (Tymoshenko) case ...is a test of credibility for the present Ukrainian authorities and that the reputation of Ukraine would suffer dramatically if it does not find a solution that we deem civilised before the European championship," Polish Prime Minister Donald Tusk told a news conference.
"I think the calls for a boycott are inappropriate. I understand the politicians who sympathise with Yulia Tymoshenko, but nothing stands in their way to express this sympathy in a clear way during the championships," Tusk said.
"I am convinced that Yulia Tymoshenko would not want her great effort and work (to bring the championships to Ukraine) to be wasted," said the Polish prime minister, whose country will be co-hosting the event.
Led by Germany, leaders of several EU countries have called off scheduled visits to Ukraine during Euro 2012, which starts in just over a month, in protest at Tymoshenko's treatment under Yanukovich's leadership.
Tymoshenko, 51, was jailed last October for seven years on a conviction of abuse of power as prime minister, a case described by many in Europe as politically motivated. She has been on hunger strike after saying she had been beaten by prison guards, an accusation denied by jail authorities.
On Wednesday, Polish President Bronislaw Komorowski warned against a Western boycott of Ukraine during the championships, saying this would send the former Soviet republic back into the arms of Russia.
The EU has condemned Tymoshenko's conviction as an example of selective justice and warned Ukraine that its members would not ratify agreements on political association and free trade as long as she remains in prison.