The 35-year-old's deal with Dynamo Kiev, the club where he began his career before moving to AC Milan, is due to expire in July and he has expressed interest in a move to North America's Major League Soccer.
He said he needed time to think hard about his next step and would make a decision "in a week or two".
"At the moment, I have not made a decision about my future - whether to continue playing football or retire, whether to stay in Ukraine or leave the country," he said during a visit to his old school in Kiev where he was mobbed by well-wishers.
"I have offers from clubs abroad and from Dynamo Kiev. Let me take a breather after the Euros and make a decision. I will certainly stay in football... football is a bit more than a game for me. It is what I live for," he added.
The former Chelsea forward crowned his international career with two headed goals in the co-host's 2-1 victory over Sweden this month, though Ukraine failed to progress out of the group stage.
After Sunday's quarter-final between England and Italy, which Italy won on penalties after a 0-0 draw, he congratulated his old AC Milan team mate Andrea Pirlo - who had a great game and scored with a cheeky dinked penalty.
"I think yesterday the stronger side won. Italy showed more quality than England and had more chances to score. That's why Italy deserved some luck in a penalty shootout," he said.
"It is really hard to predict. Four of the strongest teams have made their way to the semi-final," he said of holders Spain, Portugal, Germany and Italy.
"They are all well prepared physically and psychologically. We can expect very interesting semi-finals and great entertainment in these games."
