West Ham boss Allardyce believes in Vaz Te
Allardyce signed Vaz Te in January and his 10 goals were crucial in getting West Ham back in the top flight.
"The question mark against Vaz is whether he can do it in the Premier League - that's his challenge, " said Allardyce.
"At Bolton, he promised he would be an outstanding talent and then it faded and went into the wilderness.
"It was a case of being overconfident in his ability. If you get overconfident, you stop learning . That journey into the wilderness seems to have done him the world of good, it seems to have given him the desire to make sure he doesn't waste his talent any more.
"He went to Greece for a year on bad money and had to fight his way back. He did a great job for Barnsley and we were very fortunate he hadn't signed a new contract with them, otherwise it would have been millions to buy him rather than hundreds of thousands.
"Hopefully we will see his talent on a regular basis like we did when he arrived here. His goals were the reason we ended up in the play-offs, winning promotion.
"As soon as that consistency is hit at this level, we'd hopefully sit down and discuss a new contract with him."