England face the Netherlands in Guimaraes on June 6 in the semifinals of the competition and should they win they will face Switzerland or Portugal three days later in Porto.
The decision to leave out right-back Trippier comes a year after he was one England's leading players in their run to the semifinals of the World Cup in Russia.
"Kieran Trippier has had such a big impact, he had such a brilliant World Cup. He has been a really valuable member of our squad and has a great attitude with us, real passion in playing for England but Trent [Alexander-Arnold] and Kyle [Walker] have finished the season so strongly," said England manager Gareth Southgate.
"So it is a position where we have got real competition for places. That is as hard a decision as we have had to make," he added.
Winks has only recently returned from injury and like Trippier could be involved in the Champions League final on June 2 for Spurs against Liverpool in Madrid.
Although not part of the formal 23-man group, Ward-Prowse and Redmond will remain in training with the rest of the England squad and could possibly come into contention if there are injuries.
"Some players who have had really good finishes to the season like James Ward-Prowse and Nathan Redmond are staying with us for this period to train because although we have cut the squad to 23, we have still got two weeks to get through and two really big finals ahead. So some of those calls are even more difficult," he said.
Full England squad:
Defenders: Trent Alexander-Arnold (Liverpool), Ben Chilwell (Leicester City), Joe Gomez (Liverpool), Michael Keane (Everton), Harry Maguire (Leicester), Danny Rose (Tottenham), John Stones (Manchester City), Kyle Walker (Manchester City).
Midfielders: Dele Alli (Tottenham), Ross Barkley (Chelsea), Fabian Delph (Manchester City), Eric Dier (Tottenham), Jordan Henderson (Liverpool), Jesse Lingard (Manchester United), Declan Rice (West Ham).