Rice has started all three of England's games at the World Cup in Qatar, helping the Three Lions top their group at the tournament for the first time since the 2006 edition in Germany.
The West Ham midfielder has been linked with the likes of Manchester United and Chelsea in recent years, and his solid international displays will have done little to reduce interest in his services.
Asked whether his desire to reach the top of the club game had increased as a result of his time with Gareth Southgate's side, Rice said: "Yeah, 100 per cent.
"For the last two or three years I have been saying that. I have been playing consistently well for [my] club and I feel like I really want to keep pushing.
"I see my friends here who are playing in the Champions League and for big trophies.
"You only get one career, and at the end of your career you want to look back at what you have won and the biggest games you have played in. I am really ambitious."
Rice has made seven interceptions at the World Cup so far – at least five more than any of his England team-mates, while only Jude Bellingham (18) and Luke Shaw (17) have recovered possession more often than he has for the Three Lions (14).
England takes on Senegal for a place in the quarter-finals on Monday (AEDT), having progressed through five of their previous seven last-16 ties at the World Cup (failing to do so against Argentina in 1998 and Germany in 2010).