How to redirect the output of the time command to a file in Linux?

Try

{ time sleep 1 ; } 2> time.txt

which combines the STDERR of "time" and your command into time.txt

Or use

{ time sleep 1 2> sleep.stderr ; } 2> time.txt

which puts STDERR from "sleep" into the file "sleep.stderr" and only STDERR from "time" goes into "time.txt"

Simple. The GNU time utility has an option for that.

But you have to ensure that you are not using your shell's builtin time command, at least the bash builtin does not provide that option! That's why you need to give the full path of the time utility:

/usr/bin/time -o time.txt sleep 1