The Sun says Beckham has promised to tell West Ham soon whether he will end his career at Upton Park.
Former England skipper Becks, 35, was born in East London.
The Manchester United legend hopes to make his comeback from an Achilles injury in the next few weeks then return to Europe in January.
Hammers co-owner David Gold said: "We had some discussions with David's agents a few months ago and it was a possibility.
"He's preoccupied with a number of projects at the moment but he's promised he will give every consideration to the possibility of joining us."