The 32-year-old has been without a club since his release from Galatasaray at the end of last season and would be able to sign outside of the transfer window due to his free agent status.
Eboue did not make a single first-team appearance for Galatasaray in 2014-15 and last played a competitive match in April 2014.
Allardyce, though, revealed his Premier League strugglers could swoop if the training stint to get his fitness to an acceptable level proves successful.
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"He will still be here for a few more days to see whether he can catch up with the level of fitness there needs to be," Allardyce told reporters.
"He knows he has not played for a long time and really at the moment it is more a fitness regime to get him fit enough to play than a trial.
"If he is fit enough to play, we will probably play him a couple of behind-closed-doors games and make a decision from there.
"If he has got any sort of quality left that he had and showed at Arsenal then he may be able to secure a short-term contract with us later down the line.
"If Emmanuel Eboue has still got what he had at Arsenal then he could be very important to us later on in this season."