Some linked servers are set up to only work for a given set of windows authentications.
So, the job needs to be set up to run as one of those users.
There may be some system admin tasks that have to be done to jiggle with the account settings so that will work.
It's also possible to loop thru the list of linked servers and automatically test all of them either don't require a specific login connection or which require one but you are on the list.
Don't have sql server up and running at the moment, but there are two system views, both obvious as dirt because of their names, that you can use for that purpose.