The Lions put the non-exclusive franchise tag on Avril in March to retain the defensive end with a $10.6 million, one-year deal if both sides can't agree on a multiyear deal.
"Our goal is to have a long-term deal done with Cliff Avril," Mayhew said. "We've been working toward that the entire time. I don't think the ball's in anybody's court. I think the ball's in everybody's court."
Mayhew declined to comment Wednesday about talks with Jim Schwartz. The coach also was mum about negotiations to extend his stay with the franchise beyond the one year left on his contract.
Mayhew is entering his fourth full season and 12th with the team. He said the Lions have a talented roster with a mix of young and experienced players that learned how to win during last year's run to the playoffs.
Information from The Associated Press contributed to this report.