Germany sought English backing for its own bid to host Euro 2024 and in return offered support for an English bid for the tournament four years later.
"It would be a big ask to get Euro 2024 four years after hosting the final in 2020," Clarke told reporters at the unveiling of the Euro 2020 logo at London's City Hall.
"Certainly later on in the decade, we are reaching out to our friends in UEFA, building relationships and an ability to bid if we consider it appropriate."
Clarke also sounded a note of caution, citing England's failed bid for the 2018 World Cup, which did not progress past the first round of voting.
"We have reviewed this a number of times and the board have decided we will look at bids on a case-by-case basis," he added.
"The last (World Cup bid) cost us 18 million pounds (18.03 million pounds) and we had no chance of winning."
The impact of Britain's decision to leave the European Union also came up for discussion, with FA chief executive Martin Glenn saying 'Brexit' would not adversely impact crowd numbers for the 2020 final.
"Government would not want to be seen to be the football prevention department. They are going to want foreigners to come over and feel happy, tourists to feel happy, they want to see a great tournament," he added.
"Whatever Brexit involves, it's not going to prevent tourists coming to Britain. I believe that common sense will have to break out in a massive proportion on that whole thing."
England last hosted the European Championship in 1996.
(Reporting by Nivedita Shankar in Bengaluru; Editing by John O'Brien)
