Environment Minister Norbert Roettgen told German tabloid Bild's Monday edition that the "dictatorship" of Ukraine must not be able to use the June championship to prop up its image without reforms.
Bild quoted Roettgen as saying that he thinks "visits by ministers or prime ministers are beyond question under the current circumstances."
Roettgen's remarks were confirmed to The Associated Press on Sunday by his spokeswoman Julika Lendvai. They were the first call by a top European government minister to avoid the matches.
"The Ukrainian government should immediately release Ms. Tymoshenko to freedom," he was further quoted as saying.
The Ukrainian opposition leader is serving a seven-year prison sentence on charges of abusing her powers as prime minister in negotiating an energy deal with Russia. The West has strongly condemned the case as politically motivated and threatened to freeze co-operation with Ukraine.
The environment minister — who has no direct power on foreign policy matters — is an ally of conservative Chancellor Angela Merkel, who has been very critical of Kyiv's handling of the Tymoshenko case herself.
Roettgen is currently campaigning as the conservative's top candidate in a state election due in mid-May.
Tymoshenko has been on a hunger strike for about ten days to protest alleged abuse in a Kharkiv prison. She claims that guards punched her in the stomach and twisted her arms and legs while forcibly taking her to a hospital to be treated for debilitating back pain, stemming from an untreated herniated disk.
Tymoshenko denies the abuse of power charges against her, saying they are part of a campaign by President Viktor Yanukovych, her longtime foe, to bar her from politics. Yanukovych has denied involvement in the case.
Germany has been leading the European Union's critical stance on Ukraine over the Tymoshenko case. The government is offering to treat her in Berlin, but Kyiv has rejected the offer. Tymoshenko refuses to be treated in Ukraine.
Germany's Interior Minister Hans-Peter Friedrich, also in charge of sports, told the AP in an interview Friday that a boycott of the Ukraine matches by top officials seems possible under the current circumstances, but he stopped short of endorsing the idea.
Opposition leader Sigmar Gabriel told Bild that "there can be no normal relations with Ukraine as long as people are jailed and abused in the country for political reasons."
The Social Democrat endorsed calls urging top officials to boycott the matches, "because in the stadiums, they might be sitting next to the secret police or prison directors."
Juergen Baetz can be followed on Twitter at www.twitter.com/jbaetz