"The letter is genuine. We submitted it and are his representatives. Tomorrow we will start discussions with the political parties," longtime family friend Arturs Avotins told AFP.
The application was made to parliament in an official letter dated October 31 and addressed to four of the five parties, plus six independent MPs.
The letter was not addressed to the opposition Harmony Centre party, which is seen as pro-Moscow and has a cooperation agreement with Putin's United Russia party.
Latvia overhauled its laws this year to make dual citizenship possible. Russia was left off the list but parliament can make exceptions for individuals of "special merit" or who have been of service to Latvia.
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But Kasparov, 50, also reportedly intends to keep his Russian passport.
AFP was not immediately able to confirm the application with Kasparov, who is in voluntary exile outside of Russia, possibly in the United States or Switzerland, fearing arrest.
He has long been a fierce critic of Russian President Vladimir Putin, calling him a "dictator".
In the letter, Kasparov recalls winning chess tournaments in Latvia as a junior player and promises to promote chess in the Baltic state if his application is accepted.
Yulia, the mother of his son Vadim, also hails from Latvia.
Kasparov, who was world chess champion from 1985 to 2000, has announced he will run for president of the International Chess Federation next year.