However, according to David Ornstein, the Hammers could be in line for a reprieve should the Blues replace manager Frank Lampard.
The Athletic recently claimed those behind the scenes at Stamford Bridge had begun to look for potential replacements for Lampard amid a run of poor results.
Now, speaking to the publication's YouTube channel, Ornstein said he suspected Chelsea's interest could wane were they to sack him.
"I suspect they wouldn't," he said when asked if the urge to sign him would continue in that event from the 02:59 mark onwards.
"This is a target that has been driven by Lampard and, as I've reported in the past, there are some at Stamford Bridge who harbour reservations.
"Including whether Rice has corrected certain perceived weaknesses that contributed to the decision to release him from their academy as a teenager and also the sheer scale of finance it would take."
GIVEMESPORT's Jonathan Gorrie says...
Chelsea's potential loss could be West Ham's gain once again.
Already having made the most of the Blues' decision to release him, both he and his current club have benefitted. Indeed, Rice has established himself as a key player in East London - averaging the most passes per game of anyone in the side (48.2, via WhoScored) - as well as proving a key cog in Gareth Southgate's England team.
West Ham, meanwhile, have a potential leader for the future with so much of his career ahead of him.
While interest may yet still emerge from either Chelsea or other major sides (Manchester United have been linked before) the idea of seeing at least some interest subside has to be taken as good news.News Now - Sport News