The surprise omissions are further signs of England's disastrous Ashes tour of Australia this summer, which has led to the departures of head coach Chris Silverwood, assistant coach Graham Thorpe and team director Ashley Giles.
Anderson, 39, is England's all-time Test wicket-taker, having captured 640 scalps in 169 matches, while the 35-year-old Broad is second on the list with 537 wickets from 152 appearances.
BBC cricket correspondent Jonathan Agnew says he doesn't think the duo are "very happy" about it, but he also believes there's "a temporary feel".
Interim managing director Andrew Strauss says the non-selections of Anderson and Broad doesn't spell the ends of their Test careers.
"We feel that it is important to look at some exciting new bowling potential and give some added responsibility to other players who have featured previously," Strauss said.
"No one doubts the quality and experience that James and Stuart bring to the England set-up.
"It will be up to the new managing director and permanent head coach to decide on whether they will be involved this summer and beyond.
"This selection of this squad is the start of a process and a journey to get England Test cricket back to where it needs to be - and the hard work starts now."
Paul Collingwood was last week appointed as England's interim head coach, and the batting great, Strauss and head scout James Taylor picked the 16-man squad for the West Indies tour, which will see the sides meet in three Tests.
England have also dropped Dom Bess, Sam Billings, Rory Burns, Jos Buttler, Haseeb Hameed and Dawid Malan, who'd all played in the Ashes.
"With the start of a new cycle, it has allowed the selection panel to refresh the Test squad with a particular focus on competing away from home," Strauss said.
"We felt that it was time to draw a line after the Ashes defeat, look forward and give some impetus with an influx of new players."
Agnew labelled the omissions of Anderson and Broad "quite dramatic stuff".
"They are ageing, but I don't think they are very happy about it," Agnew said.
"They certainly don't think this is the end of their careers.
"It has to come to an end sometime and I think there is the view that this tour is a bit of a holding pattern; there is a very temporary feel about it.
"There is a huge amount of flux around English cricket at the moment.
"It may turn out this trip will turn into a hiding to nothing. They lost last time they went, so it's an opportunity to look at what there is out there."
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