The Home Office minister Lord Bassam of Brighton said there had been a sharp rise in the number of asylum applications, from 55 in January to 150 in June. Many of them were Roma gypsies who claimed to be fleeing persecution.
Lord Bassam said that although they might well be facing discrimination in their home country, they did not fill the criteria for refugees.
So far this year, all the applications from Roma people had been rejected, although a couple had been allowed on appeal, he said.
Lord Bassam said the Government was reluctant to impose visa restrictions on all Czechs.
"We would prefer not to take that option. If the situation does get out of control, that is something we will have to very seriously consider. We don't want to undermine our economic relationship with the Czech Republic, which I'm sure is very beneficial to both them and us."
Discussions have been held with the Czech authorities over the position, involving the British ambassador in Prague and government ministers.
Czech officials said any imposition of visa requirements would be a "backwards step" for the country's development as a democratic nation with close links with the EU.
The latest figures show that 192 people from the Czech Republic claimed asylum in Britain last month. In January of last year there were just five applications.