The French striker has also been linked with Manchester United, Inter Milan and Real Madrid in recent months.
But Benzema, one of the hottest properties in European football wants to stay at Lyon.
“We’ve tried to sign him several times but it’s never been possible,” Begiristain told the official website of the French football league.
“We tried this summer, but he wanted to stay at Lyon.”
Barca remains interested in the 20-year-old striker.
“In the end it will be his decision,” added Begiristain.
“We have excellent relations with both him and his club and we’ll definitely keep in contact so that he knows we want him, and that he wants to come to us.”