Hodgson has delayed the announcement of his squad to take to France until after the final Premier League fixtures this weekend, with the Football Association citing “fitness reasons”.
He has already lost Arsenal’s Alex Oxlaide-Chamberlain and Danny Wellbeck to injury, with Manchester United defender Luke Shaw another to miss out.
Former West Ham and Tottenham striker Defoe said he joined Sunderland last January from Major League Soccer side Toronto, with an England place at the back of his mind.
Defoe, capped 55 times by England, said: “As a forward you get judged on scoring goals. The England managers at the time thought I could always come on and score. When I came back I thought ‘all I’ve got to do is go there, focus, get my goals and hopefully get in the frame and take it from there’.”
Tottenham’s Harry Kane (25) and Leicester’s Jamie Vardy (24) are the only Englishmen with more top-flight goals this season and Sunderland manager Sam Allardyce said his striker could “still do it at the highest level”.
Defoe’s last cap was in November 2013 and he has previously said he would “never be at peace” with missing out on the 2014 World Cup squad for Brazil.