Fekir has scored six goals in Ligue 1 this season to help Lyon climb to third in the table and Fournier wants to keep hold of the midfielder.
Fournier also has concerns over his attacking options in light of Benzia's ankle sprain, sustained in Wednesday's Coupe de la Ligue loss to Monaco, and the imminent departure of Clinton Njie and Mohamed Yattara to the Africa Cup of Nations.
Regarding Fekir, Fournier said in quotes reported by L'Equipe: "I think it would be normal to upgrade his contract. There would be consistency and equity to do so, from what he showed on the field."
And on the matter of strengthening his squad in January, the 47-year-old added: "Yassine suffered from a severe ankle sprain that will [keep him out] for four to six weeks.
"I was not [thinking about] the arrival of a striker in the transfer window because I just did not want to pull the rug from under the feet of Yassine Benzia as he returned to a good level.
"Now we must think about an offensive player, but we are not the only club looking, nor the richest. And in our position, we should be demanding on the quality of the player."
Despite their midweek cup setback, when Monaco won 5-4 on penalties, Lyon are enjoying an excellent campaign, particularly at home.
In 10 league games at the Stade de Gerland, Lyon have won nine and scored 25 goals, conceding just five.
Lyon sit only two points adrift of leaders Marseille, with a trip to Bordeaux to come on Sunday before a two-week winter break.
