Fulham are leading Premiership rivals Southampton and Manchester City in a £1million chase for the Elland Road skipper.
The Cottagers have made an initial approach for Matteo, 30, and will follow up with a firm bid.
Matteo, a former Scottish international defender, moved to Elland Road in a £4.75m switch from Liverpool four years ago.
Leeds need to slash their £38m-a-year wage bill after dropping into Division One and Matteo is one of their big earners on £25,000 a week.
Leeds United have already rejected a season-long loan offer from Manchester City for Matteo.
The relegated west Yorkshire outfit are off-loading all their big earners, but would prefer the cash instead to ease their massive debts.
Manager Kevin Keegan has no money at his disposal – not even the £1million it would take to buy the 30-year-old Scot.
