The highly-rated duo have both signed pre-contract agreements with the club and will join them in the summer.
Dzemaili, 20, is already captain of Zurich and featured in Switzerland's World Cup squad last summer as well as the friendly against Germany on Wednesday.
He has agreed a two-year contract while 23-year-old Cid, a former France Under-19 international, has committed to four years.
Bolton boss Sam Allardyce told the club's official website: "Blerim is a fantastic player, who has played for the best team in Switzerland for the past three years. For someone so young, he has a wealth of experience.
"At the tender age of 20, he is the skipper of FC Zurich and is expected to become a regular international for Switzerland in the years ahead.
"He has hardly missed a game for FC Zurich since he started playing for them at the age of 17. I am excited by the prospect of working with Blerim next season."
He added: "We have been monitoring Gerald for quite some time and our scouts have rated him as one of the best young defenders in France."