The 31-year-old, who won the Premiership title with Saracens last month, has signed a three-year deal with the French side and said he wanted to concentrate on his new club challenge.
Under RFU rules, Lancaster cannot select players plying their trade for a club overseas unless in "exceptional circumstances" such as a long injury list.
"The opportunity to join Clermont and the subsequent decision to leave Saracens obviously was difficult and not one I have taken lightly," Strettle, who has played 14 times for England, said in a statement.
"Having now committed to a three-year move at what I feel is the right time in my career and for my family, I would like to begin that challenge right away.
"With the England training camp beginning this week I wanted to be fair to Stuart and the rest of the squad before that started."
Lancaster said Strettle's decision was "disappointing" and said Bath Rugby's Semesa Rokoduguni has been called up.
Lancaster said: "Clearly a lot has been going on with David recently with the late interest from Clermont and his decision to leave Saracens," he said.
"After visiting France recently and discussing it with his family he has made the decision to move early to get settled before the season in France starts.
"Obviously it is disappointing but David and I have chatted and he feels that, having made the move, he wants to get on with forging his new career and we both agreed it was the best course of action for him to withdraw from the squad."
Rokoduguni, a serving soldier with the Royal Scots Dragoon Guards, made his first England appearance against New Zealand in November 2014.
"I'm delighted to have been given the opportunity to join England for their World Cup training camp," he said. "It's a huge privilege and I'm really looking forward to meeting up with the boys."
(Reporting by Martyn Herman, editing by Pritha Sarkar)
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