Edelstein's announcement marks the official start of the race for the position, which has been held by Peres since 2007.
Candidates must receive the backing of at least 10 other MPs, and formally present their requests in writing to the speaker on May 27, Edelstein said.
In Israel, the post of president is largely ceremonial and executive power rests with the prime minister.
Peres will step down at the end of July, just before his 91st birthday, following a political career spanning nearly seven decades.
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"Peres is one of a kind - he was the director-general of the defence ministry in 1950s and has ever since been in every junction of decision making... So it's almost impossible to aspire to continue in his footsteps," Plesner told journalists.
"A president like Peres has a strong persona and the ability to set the agenda on issues that are important. In this respect (the presidency) is a kind of soft power and depends on the persona of the president."
Edelstein's announcement came after months of vigorous unofficial lobbying by would-be candidates, among them Energy Minister Silvan Shalom.
Shalom, a member of Prime Minister Benjamin Netanyahu's ruling rightwing Likud party, has yet to announce whether he will still run for the post.
Another presidential hopeful is veteran Labour parliamentarian Benjamin Ben Eliezer, who has held several ministerial portfolios and has already accrued the required support from MPs.
Also today, former Knesset speaker Reuven Rivlin of Likud formally announced his candidacy. He also ran in the 2007 race, losing to Peres.