Current stopper Mark Schwarzer is expected to join Arsenal before the season starts in a £3.5m deal and when the Australian completes his move, the West Londoners want Green as a replacement.
The West Ham goalkeeper had been considered for the club most of the summer whilst club officials had also been eyeing Glasgow Rangers 'keeper Alan McGregor.
They also quietly hoped to be able to snare David James on a free transfer, however he was recently snapped up by Championship club Bristol City.