The Nigerian ace is currently plying his trade with the Seattle Sounders in the MLS, and could be available for a temporary move next month with the American season now finished.
Allardyce has been linked with a number of players of late as he plots to address attacking concerns at the club.
Andy Carroll’s injury woes have seen the Hammers forced to play a ‘strikerless’ system, which has contributed to their meagre 13 goal tally this after 15 Premier League games.
‘Big Sam’ is keen to land a player with a proven record for finding the back of the net and experience in English football, which are two requirements that Martins fills.
Seattle may grant their star striker permission to pursue a temporary move, with the 29-year-old still hopeful of securing a spot in his national side’s squad ahead of the 2014 World Cup – despite not having worn the green jersey for some time.
Martins may be one of a number of MLS players to move to England in January, with Clint Dempsey – also of the Sounders – and Landon Donovan having been mooted as targets for the likes of Fulham and Everton.