Qatar is expected to unveil a tender to replace its aging fighter fleet of nine Dassault Mirage M-2000ED ground-attack jets and three Mirage M2000-Ds, some of which were deployed on NATO-led operations in Libya during the 2011 war that toppled Col. Moammar Gadhafi.
The Qatari air force also has a squadron of Alpha jets.
France is desperate to secure foreign orders for the Rafale, and Hollande, who was in Qatar to attend a meeting of Western and Arab states that support rebel forces in Syria's 27-month-old civil war, has sought to capitalize on France's close defense and commercial links with the tiny gulf state.
The multi-role Rafale, which so far is only in service with the French air force and navy, is in competition with Boeing's F-15 Eagle and the Eurofighter Typhoon manufactured by European defense titan EADS and marketed by Britain's BAE Systems.
Holland was accompanied by French defense sector officials, including Eric Trappier, Dassault's chief executive, who said the Rafale "interests the Qatari air force."
The Indian air force chose the agile, twin-engine Rafale in January 2012 when it ordered 126 in a deal worth $15 billion.
The French jet, which entered service in 2006, has become the standard bearer of the French aerospace industry.
It beat out the Eurofighter Typhoon in the final round in India. Lockheed Martin's F-16 and Boeing's F/A-18 had been eliminated earlier.
But that deal still has to be finalized by New Delhi. If it is, it will be the Rafale's first overseas sale.
The fourth generation combat aircraft has seen action with French forces in Libya and Mali, but is generally viewed as lagging behind fifth generation multirole fighters like Lockheed Martin's F-35 Joint Strike Fighter.
Dassault has been negotiating with another gulf state, the United Arab Emirates, since 2008 to sell 60 Rafales potentially worth $10 billion to replace the Mirage 2000-9s it bought more than a decade ago.
But negotiations went into a tailspin in November 2011, when the Emirates, the world's fourth largest oil exporter, ruled the proposed terms were "uncompetitive and unworkable."
The crown prince, Sheik Mohammed bin Zayed al-Nahyan, who is also deputy commander of the Emirates' armed forces, then sought details of the Typhoon, of which Saudi Arabia bought 72 in 2009 for $10 billion.
The Emirates wants the Rafale's Safran M88 engines to be upgraded with extra power -- 9 tons of thrust, 1.5 tons more than the variants operating with the French air force.
The Emirates also wants more advanced Thales radars, electronic warfare systems and avionics installed, industry sources have said.
Kuwait turned down Rafale to replace its aging Mirages several years ago as too expensive following pressure from Parliament.
Gulf military sources said the Rafale's prospects in Qatar will be greatly enhanced if the deal with the United Arab Emirates finally goes through, as this would facilitate joint operations and use of facilities by the two states.
Holland visited Abu Dhabi, in January but no breakthrough was announced.
In December, the French Defense Ministry warned Dassault would halt production of the Rafale in 2021 if it did not win any export orders.
So Hollande, battling for key defense exports to keep French assembly lines in business amid military spending cutbacks, has a lot riding on the jet in his sales drives in the Persian Gulf.
The French government has been dismayed by a string of failed efforts to sell Rafale to South Korea, the Morocco and Brazil, with the final word on the all-important Indian order still hanging.
Competition, even within Europe, has become intense as Europe's combat aircraft industry faces an uncertain future.
"This leaves Europe's two main military powers, France and the United Kingdom, continuing to grapple with the problem of sustaining their respective combat aircraft sectors in the medium to long term," the International Institute for Strategic Studies in London observed in a recent analysis.