The verb surveil, originally a backformation of surveillance, was long considered nonstandard, and even now is still so new to the language (the earliest instances date from the early 1960s)1 that some dictionaries don’t include it, and your spell check might disapprove of it. But even though survey is closely related, etymologically, to surveillance, survey does not carry the sense to keep under surveillance (where surveillance means close observation, especially of one under suspicion).2 For this purpose surveil works better, so the word is a useful addition to the language.
Surveil‘s participles are surveilling and surveilled. Again, your spell check might not like these words, but they’re fine.
It doesn’t put a microphone in every government office or surveil every meeting between lawmakers and public employees. [Carolina Journal]
An unlucky few will be stalked, fired, surveilled, arrested, deported or even tortured. [Wall Street Journal]
Fowlkes believed his wife was having an affair and had been privately surveilling her to prove it, sources said. [New York Daily News]