Clarke, who last month succeeded Greg Dyke as head of the FA, said he began thinking about the country's chances of hosting a major international tournament after England's 1-0 World Cup qualifying win against Slovakia in Trnava on Sunday.
"I was mulling on the way back on the plane from Slovakia what Brexit does to our chances of a future tournament because I never had any expectations of a new tournament (bid) in my tenure," Clarke was quoted as saying by British media on Wednesday.
"The timescales are too long but you have to build a platform for the person after you or the person after them to be able to harvest some goodwill and get a tournament because this country deserves a tournament.
"But we are leaving the EU. Can we count on EU support or will they vote for EU nations in the future? I don’t know the answer to that but we have to get to the bottom of that."
England have been linked with bids for the 2028 European Championship and 2030 World Cup in media reports, but Clarke said his priority was forging independent relationships European soccer's governing body UEFA and its world counterpart FIFA.
"We have to build relations directly within UEFA now and not rely on the support of the EU nations because they might say we've got an EU member who wants it (a major tournament) this year," he added.
"So it's doubly important to have really strong relationships in UEFA and FIFA and I'm going to be working hard at that."
The FA boss also said he was open to the idea of England co-hosting tournaments with EU nations.
"I need to find out before we waste a lot of money," he said. "It could mean we have to slightly change our tactics for getting a tournament. It might be jointly hosted with an EU member."
England, which hosted the European Championship in 1996 and the World Cup in 1966, will stage the semi-finals and final of the next European championship at London's Wembley Stadium, when the tournament is hosted across the continent in 2020.
(Reporting by Ian Rodricks in Bengaluru; editing by Ken Ferris)
