While Venezuela's big banks are generally seen as structurally sound, a number of takeovers of small failing banks since last year triggered rumors of impending disaster in the sector, worrying savers.
Anybody who spreads malicious rumors by any means, emails, SMS messages on cellphones, Twitter, Facebook ... or any other means of communication is committing a crime and should answer for it before the competent authorities, Wilmer Flores Trosel, head of the CICPC national investigative police unit, said in a statement.
He said a man was arrested in possession of a cellphone used to send the first illegal Tweet, and that detectives had also detained a woman and seized hard discs and computer equipment from her.
Both suspects were picked up in the eastern city of Ciudad Bolivar, CICPC said, following a four-month investigation into the posting of false information about the nation's financial system on Twitter with the aim of creating runs on banks.
President Hugo Chavez's government closed 11 small banks last year that had been run into the ground by businessmen with links to public officials.
Then last month it took over the mid-sized Banco Federal, which had close ties to the opposition Globovision TV station, citing liquidity problems and risk of fraud.
Every move Chavez's government makes is commented on and critiqued, often sharply, by local Twitterers.
The social networking site's popularity has exploded in Venezuela this year and Chavez has joined in. His @chavezcandanga account was launched in April and quickly overtook the Globovision feed to become the country's most closely watched with more than 645,000 followers.
(Editing by Frank Jack Daniel and Kieran Murray)