LONDON — Britain’s domestic intelligence chief has demanded greater authority for spies to help fight the threat of Islamist extremism, a sign that the attack on a satirical newspaper in Paris is likely to sharpen the security-versus-privacy debate in Western countries.

Andrew Parker, the director general of MI5, said militants were planning attacks in Britain similar to the one that killed 12 people at the newspaper, Charlie Hebdo.

More than 600 Britons have traveled to Syria to join jihadists, he said, and three terrorist plots in Britain have been stopped by security services in recent months alone.

“Death would certainly have resulted otherwise,” he said.

Amid a backlash against digital surveillance after disclosures by the former National Security Agency contractor Edward J. Snowden in 2013, Mr. Parker said there was a growing imbalance between the number of terrorist plots against Britain and the ability of spies to track their communications.