Jonathan de Guzman says he turned down QPR to join Napoli, despite the Premier League club offering a better deal.
Chelsea goalkeeper Petr Cech has reacted positively to being dropped, says manager Jose Mourinho.
Cole was released by the Hammers in the summer after making more than 200 appearances over seven years.
But Allardyce has been desperate to bolster his attacking options, with Andy Carroll injured, and will return to the 29-year-old once he proves his fitness.
"We've had Carlton training with us for a while now and as soon as he's fit enough to play there will be an offer of a short-term contract on the table,
there's no doubt about that," Allardyce said.
"That's why he's here with us and that's why we're trying to improve his fitness having missed pre-season.
"As soon as he's okay and suitable terms and conditions have been agreed, which is nothing do with me, then he'll join us for however long the contract is negotiated for.
"We'll play a couple of behind-closed-doors games just to let him get some match practice and then we'll take it from there."
West Ham have not won in the league since the opening day and Allardyce expects to face a Hull side high on confidence when the teams meet at the KC Stadium on Saturday.
Steve Bruce's newly-promoted side twice came from behind against Newcastle to claim their first win of the season and go into the encounter two points ahead of the Londoners in the table.
"For them to get their first away win was a huge result, it lifts the whole city, the backroom staff and all the players," Allardyce added.
"We face them at a time when their confidence is sky-high so as always we'll have a tough game on our hands.
"His (Bruce's) players, irrespective of how many he's changed, have gelled well, which was the opposite to his time at Sunderland.
"At Hull he's changed a lot of players and it's worked for him very, very well in a short period of time and they've obviously started the season exceptionally well."