Chelsea made attempts to sign Rice when Frank Lampard was manager and even though there has been a change of ownership and management at the club, the 23-year-old midfielder remains a key target for Chelsea.
West Ham have been clear that they are not ready to sell their captain, but Rice fuelled rumours over his future with comments made in an England press conference on Thursday.
"I've been playing consistently well for the club and I want to keep pushing," said Rice.
"I see my friends here who are playing Champions League football, winning the trophies.
"You only get one career and at the end of your career you want to look back at what you've won and the biggest games you've played in."