If Rouhani accepts, he will be the first Iranian president to make a state visit to France since 1999.
Fabius is seeking to smooth over any strains in the relationship created by France's hard line during negotiations that produced a historic nuclear agreement on July 14, and thereby create a climate conducive to stronger economic ties.
France hopes to secure business in Iran once Western sanctions are lifted under the nuclear deal, and Fabius said last week that his tough stance in last month's negotiations would not stand in the way of French commercial opportunities.
"Things will, we hope, be able to change," he told reporters in Tehran in comments broadcast on French television. "On both sides we want to develop our ties to set a new direction," he said, confirming Hollande's invitation to Rouhani.
The French minister met his Iranian counterpart Mohammad Javad Zarif and Oil Minister Bijan Zanganeh on his one-day visit and was due to meet Industry Minister Mohammad Reza Nematzadeh, and Masoumeh Ebtekar, head of the environmental Protection Agency, Iran's state news agency IRNA said.
Zarif said that the meeting had gone well. "We began good discussions to reach regional cooperation in the fight against terrorism and extremism," he said in a televised news conference.