While Ireland play their three group games in the Polish cities of Poznan and Gdansk, Irish fans could have to travel to the Ukraine if they qualify for the quarter finals. England play all their first round matches in Ukraine.
The advice came after Kiev's first ever gay pride parade was cancelled on Sunday amid fears of violence from far-right thugs.
Television pictures showed Svyatoslav Sheremet, head of the Gay Forum of Ukraine, being kicked and jumped on by a group of men after the event was stopped.
Amnesty International said police in the capital advised organisers to abandon the march just 30 minutes before it was due to start after 500 ultra-right football hooligans had gathered.
Thousands of England fans will travel to the eastern European country for Euro 2012, which begins on June 8.
Amnesty International Ukraine campaigner Max Tucker said: "Gay football fans will have extremely good reason to be concerned.
"Not only will they have to deal with violent football hooligans who deliberately target gay people and people of ethnic minorities, they will also face an extremely corrupt police force who have a track record of beating and mistreating people because of their sexual orientation."
He added: "I would advise any gay football supporters going to the Ukraine to go there with extreme caution and be on the lookout for both the police and hooligans and try to keep as low a profile as possible."
The British Foreign and Commonwealth Office (FCO) issued a travel warning which said conditions for gay people in Ukraine were deteriorating.
It stated: "Although homosexuality is legal in Ukraine, public attitudes are less tolerant than in the UK and public displays of affection may attract negative attention.
"There is no provision under Ukrainian legislation guaranteeing freedom from discrimination on the grounds of sexual orientation and there has recently been an increase in hostility towards the LGBT community."
Gay rights campaign group Stonewall called on the UK authorities to provide reassurance over fans' safety.
Sam Dick, head of policy at the organisation, said: "The shocking violence against gay people in Ukraine will be of considerable further concern to the many lesbian and gay fans heading to watch Euro 2012.
"We expect both the Football Association (FA) and the Foreign and Commonwealth Office to answer the serious questions about what they have done to make sure gay people visiting and living in the Ukraine are safe during the tournament and beyond."
Last week the brother of black England midfielder Theo Walcott tweeted that he and his father will not be flying out to support the star because of "possible racist attacks".
Ashley Walcott, Theo's older brother, posted to his Twitter page: "Unfortunately my dad and I have taken the decision not to travel to the Ukraine because of the fear of possible racist attacks confrontations.
"Something's aren't worth risking, but begs the question why hold a competition of this magnitude in a place that can not police itself for foreigners of any creed to feel safe, but I'll be watching every minute."