Time Machine saying not enough space to work - even though there is space

Hi guys - pls help. So my macOS Monterey has 250GB of storage- I have used 246GB of this and want to back up my data using time machine to a hard drive (My Passport) which has 1TB of space. It mounts to my laptop and I erased the drive- it now has the TimeMachine symbol but it will not go through with the backup. The error message says that it can't back up because the disk Macintosh HD is nearly full. To clarify - do I need existing space on my laptop to be able to back it up to a harddrive?? help would be appreciated.