Rodriguez joined the Dolphins in June having been released by the Chicago Bears for off-field issues after just one year with the franchise.
The 24-year-old took to Twitter to reveal the news and seemed upbeat about his future chances in the NFL.
"Time to find a new home! It's just the nature of the business. Thank you family and fans for the love and support! God always has a plan!" Rodriguez wrote on the social networking site.
Meanwhile, 28-year-old Clutts will join the Dolphins having spent last season with the Houston Texans and the campaign before that with the Chicago Bears.