Rice joined Arsenal for a club-record £100million in July, having capped his final season with the Irons by lifting the Europa Conference League trophy in June.
Despite the windfall generated by Rice's sale, West Ham endured a frustrating pre-season amid reports boss David Moyes did not see eye-to-eye with new technical director Tim Steidten.
However, the signings of former Southampton skipper James Ward-Prowse and Mexico international Edson Alvarez have lifted the mood at the London Stadium.
Ward-Prowse enjoyed a dream debut as West Ham beat Chelsea 3-1 on Sunday, becoming the first player to record multiple assists on his first Premier League outing for the club.
With Ward-Prowse bringing his set-piece prowess to the London Stadium, Redknapp – who played for West Ham between 1965 and 1972 and managed the club between 1994 and 2001 – is positive about the Hammers' prospects.
Asked about West Ham's recruitment, Redknapp told Stats Perform: "I love Ward-Prowse. He's a good player and looks a good character and a good lad.
"So yeah, they could be a stronger team this year, with the money they've raised. They've lost one outstanding player but they've [potentially] got [some] very good players in with that money.
"I think it's good business for West Ham. I think West Ham will be fine."
Asked about Moyes facing criticism from supporters despite lifting the first major trophy of his managerial career last term, Redknapp added: "That's football, isn't it? That's the way the game goes. It's a results business now, you don't get long.
"You know, back in the day, West Ham had four managers in something like 75 years. Ron Greenwood, John Lyall, Ted Fenton before that. That was how the game was.
"Now, you get 20-odd minutes! You lose a few games, you lose four, five, six games, you're in trouble. You've got people calling for your head.
"David is a good manager. He's been around long enough, he knows his situation, he knows he needs results. There's no reason they can't have a good season."
Another of Redknapp's former clubs did make a coaching change ahead of the new season, with Bournemouth replacing now-Wolves boss Gary O'Neil with Andoni Iraola.
Asked about his first impressions of the former Rayo Vallecano coach, Redknapp said: "I don't know much about him at the moment, but they tell me he's very good.
"It's a good club, with new owners, and they're ambitious. It's good that new people have come in, they look like they're good for the club. I think they're going to be good for everybody.
"It's their choice to change manager, it's their decision. They own the club, they can do what they want and they feel the new guy's the man to take the club forward. Let's hope he can.
"Things are looking good down there, I think the team looks okay. They've made one or two signings and they've got a couple more injured still to come in.
"I think they might make one or two more signings. I don't see them being a relegation outfit this year, I think they'll be fine. They'll improve on what they did last year possibly."
Elsewhere, Redknapp is pleased to see former Bournemouth boss Eddie Howe succeeding with Newcastle United, having earned his opportunity by leading the Cherries to the top flight for the first time in 2015.
"He's a top manager," Redknapp said of Howe. "We've got some fantastic young managers in this country who don't get the opportunity to manage a top team.
"There are loads in the Championship and below who could do a great job in the Premier League if they were ever given the opportunity.
"Eddie deserved the opportunity. What he did at Bournemouth was amazing, and now he's doing magnificent, wonderful things at Newcastle.
"He's got the backing, he's got good players, they've bought some fantastic players in, the recruitment's been great. But Eddie's a top manager, he could manage anywhere."
