David Moyes’ men welcome basement boys Southampton to London Stadium on Sunday (2pm), sitting in the drop zone themselves, but with only four points separating the bottom nine teams.
The Hammers have games in on hand on all of their rivals and entertain fifth-placed Newcastle United next Wednesday (April 5).
And Rice was one of 10 senior players to go away on international duty, opening the scoring for Gareth Southgate’s England against Italy in Naples with his third goal for his country.
The 24-year-old reacted quickest in the box to fire home left-footed when Harry Kane’s effort was blocked, then saw the Tottenham striker convert from the penalty spot just before the break to become England’s all-time leading goalscorer with 54.
“Every time you score is such a proud moment, let alone when you do it for your country,” Rice told Channel 4 Sport.
“I think I was shocked it dropped to me, but I just wanted to make a good contact with the ball. I think I scuffed it, but it went in and they all count.
“I was absolutely buzzing to get that goal and for Harry to then score, it topped off a great first-half performance. We really dominated and showed what we’re about and the next step is to apply that across all 90 minutes.”
Italy hit back after the restart through debutant Mateo Retegui after a mistake by Harry Maguire, but could not make their extra man count after Luke Shaw was dismissed in the 80th minute, following two bookings in quick succession.
“It was a story of two halves,” added Rice. “We completely controlled and dominated the game in the first half and could have been three goals up.
“In the second half, we put ourselves under all sorts of pressure straight from the start. But we showed all sorts of desire and the mentality needed to dig in after going down to 10 men.
“It’s a first win for us in Italy for a long time [since 1961] so it was a big night for us.
“Harry said a few words in the changing room after the game and you can see how much it [the goal record] means to him.
“Every day, ever since I walked through the door with England, the way he approaches training and scores goals in training and games, he’s a real leader for us.
“To get to 54 goals is some statement and he could go all the way to 100 if he wanted to. He’s a top player and it’s thoroughly deserved.”
Rice, who was booked against the Italians, won his 41st cap and played another full 90 minutes – his 46th game of the season – as England beat Ukraine 2-0 at Wembley Stadium on Sunday.
Kane opened the scoring from close range, with Arsenal’s Bukayo Saka adding a superb seconds moments later to put England in pole position in group C ahead of June fixtures against Malta and North Macedonia.
But Rice and West Ham face another 12 Premier League fixtures before then, with the club’s top-flight survival in the balance.
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