Thousands of Pakistani Islamists had clashed with police earlier this week in protest against the arrest of their leader ahead of rallies denouncing French cartoons depicting the Prophet Mohammad.
For Muslims, depictions of the Prophet are blasphemous.
The diplomatic sources said that a message had been sent overnight to French citizens and companies following threats by hardline Islamist group Tehrik-i-Labaik Pakistan (TLP) to target French interests.
The sources said the embassy sent a message to French residents in Pakistan recommending that French nationals leave the country and French companies shut down activities temporarily "due to the serious threats to French interests in Pakistan".
Relations between Paris and Islamabad have worsened since the end of last year after President Emmanuel Macron paid tribute to a French history teacher who was beheaded by an 18-year-old man of Chechen origin for showing cartoons of the Prophet in a class on freedom of speech.