"If support [in parliament] can indeed be found, we will assume our responsibility to the French people," Chenu told France 2 broadcaster.
He added that his party could receive support from lawmakers from other parties, who "might want the National Assembly not to be blocked."
Bardella announced before the first round of elections that he would not become prime minister unless his party and its allies won an absolute majority of seats, that is, 289 seats.
In France, if the opposition party wins an absolute majority of seats, the president is required to appoint the leader of the opposition as prime minister. If a party has only a relative majority, the president can choose any prime minister.
The National Rally and its allies obtained 33.15% of the vote in the elections, which would give it up to 270 of the 577 seats in parliament. The left-wing New Popular Front secured 27.99%, while Ensemble — French President Emmanuel Macron's coalition — gained 20.04%. The second round of elections will be held on July 7.