Rice has established himself as a top Premier League star who has helped the Hammers to one of their most memorable campaigns in years.
They reached the semi-final of the Europa League and can still finish sixth in the table.
talkSPORT understands West Ham have offered the midfielder an eight-year contract worth £200,000-a-week, but that the 23-year-old has no intention to sign it.
Rice still has two years left on his current deal, but the Hammers face the very real possibility of losing him in the near future.
"I'm hopeful [he will stay] but not necessarily confident," Hurst said in an exclusive interview with talkSPORT.
"It is a big issue for a young player with his outstanding talent at a club not in the top four or five.
"They've offered him a big contract and I'd like to see him at the club and stay.
"He's absolutely brilliant, a decent lad and fantastic player and still to improve.
"[But]If you were to get £100million plus, which is a possibility, then you could strengthen the club with three or four players in different positions."
England World Cup winner Hurst, who played 500 times for the Hammers, also revealed that he spoke to Rice over Zoom and urged him to become more of a goalscoring threat.
He added: "I did a private Zoom call to him and said to him what I'd like to see him improving is scoring a few more goals, like Martin Peters.
"He can add a few more. Defensively, he is unbelievable. I think he's too good to be just a defensive and holding midfielder.
"He's magnificent at that, but he's got a wider scope to improve and score more goals."
Sir Geoff Hurst was helping launch The FA's support of the national Thank You Day. The FA will host a Big Jubilee Lunch at Wembley Stadium on the Platinum Jubilee weekend (Sunday June 5) for 1,000 members of the local Brent Community.