Essendon chairman Paul Little confirmed on Wednesday night the club will not appeal against last month's Federal Court verdict.
While Hird has not commented publicly, there is widespread speculation that he has broken ranks with the Bombers and will challenge Justice John Middleton's findings.
Essendon could sack Hird as a result.
ASADA announced last week that it would proceed with amended show-cause notices against 34 current and former Essendon players as soon as the Bombers and Hird decided not to appeal.
Essendon and Hird took ASADA to court, arguing the anti-doping body's joint investigation with the AFL into the club was unlawful.
But Justice Middleton made a decisive ruling against club and coach on September 19.
"The players welcome the clubs decision (not to appeal) and urge ASADA if it indeed intends to issue amended show-cause notices to do so as a matter of urgency," players association chief executive Paul Marsh said.
"This matter has dragged on for far too long.
"It is essential for all parties involved to take steps to expedite the process and bring it to a conclusion.
"We do not believe that an appeal by James Hird if he does lodge one should in any way disrupt the process."