Brady was suspended without pay by the NFL on Monday for the first four games of the 2015 season stemming from his role in Deflategate.
The NFLPA called for an independent arbitrator to hear the case, although the league's collective bargaining agreement stipulates that the arbitrator will be decided by commissioner Roger Goodell or a person he designates.
"Given the NFL's history of inconsistency and arbitrary decisions in disciplinary matters, it is only fair that a neutral arbitrator hear this appeal," the union said in a release. "If Ted Wells and the NFL believe, as their public comments stated, that the evidence in their report is 'direct' and 'inculpatory,' then they should be confident enough to present their case before someone who is truly independent."
Under the collective bargaining agreement, the NFL now has 10 days to name an arbitrator and schedule Brady's hearing, although that date can be pushed back if both sides agree.
Brady's agent said Monday that he would appeal the suspension. The deadline to appeal was the end of the day Thursday. The Patriots, who were fined $1 million and lost two draft picks, have until May 21 to file an appeal of their penalties to the NFL.
The punishment was announced five days after the release of the lengthy Wells report that found that Patriots personnel deliberately deflated footballs before the AFC Championship Game on Jan. 18, 2015, and that Brady "was at least generally aware" of the violations.
"With respect to your particular involvement, the report established that there is substantial and credible evidence to conclude you were at least generally aware of the actions of the Patriots' employees involved in the deflation of the footballs and that it was unlikely that their actions were done without your knowledge," NFL executive vice president Troy Vincent wrote in a letter to Brady.
The 243-page report by league-appointed attorney Ted Wells said it was "more probable than not" that Brady was aware of plans to prepare the footballs to his liking, below the NFL-mandated minimum of 12.5 pounds per square inch.
The Patriots fired back at the report Thursday, creating a website that directly challenged some of the findings.
Information from The Associated Press was used in this report.