"I think there are days ahead, but I don't think it would be right for me to speculate about football tournaments in three months' time", the minister told Sky News broadcaster.
According to Kwarteng, although everyone wants "things started" and the economy to be up and running as quickly as possible, the COVID-19 pandemic is still a serious issue.
"I think there is still a serious issue that we have with coronavirus, there is still people who are very ill", he said.
On Monday evening, Johnson told The Sun tabloid that the UK, which is scheduled to host European Championship's semi-finals and final in July, could host the entire tournament.
"If they want any other matches that they want hosted, we're certainly on for that but at the moment that's where we are with UEFA", the prime minister said.
The semi-finals and final are set to be played at Wembley stadium, in north London, after the 2020 tournament was postponed due to the COVID-19 pandemic.
Johnson unveiled last week the government's roadmap out of the lockdown in England, where all restrictions on social contact are expected to be lifted by 21 June.