The 31-year-old Australian's current £60,000 per week contract runs out at the end of the month and one-year deal worth around £30,000 per week is all that has been offered by the Hammers as they look to trim the wage bill even further to allow Gianfranco Zola to bring in more players in the close season.
Neil's agent Darren Jackson told Sky Sports News, "The last offer was declined because Lucas didn't feel it reflected his value as a player and captain."
"Negotiations have been amicable and we hope we can still reach an agreement. The door remains open for West Ham and Lucas. Obviously the longer it goes on, the more likely it's not going to happen."