The 33-year-old was released by the Hammers just days before the close of the transfer window and is currently on the lookout for a new club.
Nolan is interesting Bournemouth, while Sunderland have also emerged as shock contenders for his signature.
However, Orient might now be potential suitors, although manager Ian Hendon appeared to suggest he was just at the club to ensure he stays fit.
"It is great to have such a model professional in Kevin at the club, and I am more than happy for Kevin to come in and train with the squad," Hendon said.
"It is good for our players to be training with a vastly experienced and talented Premier League midfielder."