Alnwick, 22, was on trial at Thorp Arch after being let go by the Magpies this summer.
But he has been told he will not get a deal and has already left the club, as reported by theYorkshire Evening Post.
The stopper shot to notoriety on his Newcastle debut last season when he made a string of saves to deny Chelsea at home in the Premier League.
He went on to make seven more outings for Newcastle before the club let his contract expire this month.
Leeds require goalkeeper recruits after releasing Stuart Taylor at the end of last season - meaning Marco Silvestri has no back-up for the number one jersey.
Silvestri enjoyed an impressive debut season but new head coach Uwe Rosler is said to be keen on adding strength to the goalkeeping position.
The club are also understood to be interested in former Chelsea 'keeper Ross Turnbull, now at Barnsley.