Both Kane and Vardy are likely to be in the England squad for the 24-team tournament, which coach Roy Hodgson is scheduled to announce on Monday.
"If I do go, it'll be a dream because it will be my third tournament and playing for England is something that I've always wanted to do," Defoe told British media.
Welbeck underwent surgery on his right knee after sustaining cartilage damage in Sunday's 2-2 draw with Manchester City and faces an extended spell on the sidelines.
Defoe earned his last cap for England against Chile in November 2013, and although keen to represent his country again, the 33-year-old is relaxed about the prospect of making to the tournament.
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"I'm happy with the way I've performed this season," he added. "I've ticked every box and if I don't go then I can say at least I've done everything right."
England face Russia, Slovakia and Wales in Group B of the tournament in France starting on June 10.
(Reporting by Ian Rodricks in Bengaluru; Editing by Amlan Chakraborty)
