The 29-year-old will be out of contract at the end of the season and, at £90,000 a week is one of the highest earners at the Hammers. The club have an option to extend his deal for another two years.
“When you need to renew your contract, you must demonstrate why they should be renewing your contract," said Pellegrini.
“We know what Andy can give to our team and now he must demonstrate he is able to do it week by week. He is now in the last six months and I hope he will be able to do it."
Pellegrini made it clear he currently views Carroll as an impact substitute, with Javier Hernandez now West Ham's first-choice starting striker.
“If he can work in the normal way, of course he can be an option to start," he said. “For the moment, I think he can be a very important substitute."