The 31-year-old had a torrid time last season as Aston Villa were relegated from the Premier League and is said to be surplus to requirements at Villa Park.
The American has been at Villa for eight years after joining from Chivas USA.
Guzan has played 171 times for Villa and also has 50 caps for the USA national team.
Middlesbrough signed former Barcelona 'keeper Victor Valdes who will most probably be first choice, meaning that Guzan may have to settle for a place on the bench if he were to join Boro.