The Hammers defender, 19, has rejected two new contract offers – the last one a deal worth about £15,000 a week.
He has entered the final two years of his current £3,000-a-week deal at the club.
Rice, who does not want to leave the London Stadium, wants a £30,000-a-week deal for the first three years, rising to £35,000 a week for the second three years of the six-year contract being proposed, plus various add ons.
That deal would still be worth only about half of what some of West Ham’s other defensive signings, such as Fabian Balbuena and Issa Diop are earning.
Rice’s situation is being monitored by some of the Premier League’s elite clubs.
England coach Gareth Southgate is also on Rice’s trail. He is eligible to play for both England and the Republic of Ireland, as he was born in London but with Irish grandparents.
Rice has previously played in friendly matches and in junior age groups for the Republic but has not played in a competitive fixture.
Pablo Zabaleta this week backed Declan Rice to be a West Ham great.
“Declan is young, he still has room for improvement for sure, but you can see he has the right attitude and ability to be become one of the greatest players for West Ham,” the Argentinian said.
“He looks so mature and he has proven week in, week out that he can perform for the team, so as a team-mate for me he is one of those young players that you like to look after and all the information I can give to him I will.
“We don’t need to put so much pressure on him now, we need to help him to keep improving. It’s step by step and then let’s see how far he can go as a player.”