Real Madrid defender Sergio Ramos insists he is in no rush to sign a new contract at the Bernabeu.
Birmingham manager Gary Rowett has confirmed he wants to extend Darren Randolph's contract beyond the end of the season.
Cudicini moved to White Hart Lane in January 2009 on a free transfer after spending nine years with London rivals Chelsea.
But the 39-year-old was restricted to just 20 appearances for Spurs because of injury - including a motorcycle crash with resulted in two fractured wrists and some pelvis damage.
The Italian stopper has fallen further down the pecking order at Tottenham and has an opportunity for some regular game time in the States.
Should the move go ahead, Cudicini will join former Tottenham striker Robbie Keane at the Galaxy.