No. When a player is claimed off waivers, his existing contract transfers to the claiming team and he has no say in it. He cannot decline the claim and become a free agent instead. The only way to avoid reporting is to not play at all, which in practice means retiring or being placed on a reserve list.
How a Claim Works
When a team waives a player who is subject to waivers, every other club gets a window to claim him. If more than one team puts in a claim, the one highest in the waiver order wins. The winning team takes over the contract exactly as written, salary included, and the player joins that roster.
The Vested Veteran Exception
A player with four or more accrued seasons who is released before the trade deadline skips waivers entirely and becomes an unrestricted free agent immediately, free to sign anywhere. Released after the deadline, the same player goes through waivers like anyone else. That is why veteran releases in the first half of the season tend to turn into open market signings, while late season releases often end in a claim.
What a Player Can Actually Do
A claimed player has two realistic options: report to his new team, or step away from the game. Contracts can include clauses that affect how they are handled, but the waiver claim itself is not something the player gets to accept or reject.
More on NFL Roster Rules
How NFL waivers work · NFL waiver order · Waivers vs free agency · Waiver claims tracker