Coady was signed from Wolverhampton Wanderers although the experienced defender spent last season on loan at Everton.
"I've been massively excited all summer and it's something I was desperate to get done. I've played against the club numerous times and it's so, so hard playing against Leicester City. That's something we need to take into the season," Coady said in a statement.
Winks arrives from Tottenham Hotspur where he made over 200 appearances for the London club but spent last season on loan at Italian club Sampdoria.
"I'm looking forward to getting started, to have a full pre-season with the team and come here early to get ready for the games coming up," Winks said.
Coady played for England last year and made the World Cup squad but did not make the squad for their European championship qualifiers, while Winks has not played for his country since 2020.
Leicester were Premier League champions in 2015-16 but they were relegated at the end of last season after they finished 18th. They last played in the Championship in the 2013-14 season when they won the title to gain promotion.
The club appointed Enzo Maresca as their new manager last month after the Italian left his role as assistant coach under Pep Guardiola at treble winners Manchester City.
But the club's biggest assets were prised away with England midfielder James Maddison (26) moving to Tottenham Hotspur in a 40-million pound deal while Aston Villa signed Youri Tielemans (26) for free.
