The Frenchman has a year left on his current deal and was linked with Paris St Germain in the summer but dismissed the prospect of a move from the club he joined in October 1996 prior to the end of his existing contract.
Gunners chief executive Ivan Gazidis has stated his intent to retain Wenger beyond the end of the season, but AST wants the club to wait before determining Wenger's future after eight trophyless years and the 3-1 defeat to Aston Villa in their opening Premier League match on Saturday.
A statement from AST read: "In recent weeks Ivan Gazidis has spoken of the club's intent to offer Arsene Wenger a new contract. The AST believes this is inappropriate.
"A decision on whether to offer an extension to the manager's contract which ends in May 2014 should only be taken following the closing of the transfer window and assessment of the team's performances in the weeks ahead.
"Considerable improvement must be shown for us to be convinced this should happen, and we believe that other senior football appointments must come first.
"The failings we highlighted in Arsene Wenger holding far too much power have only been amplified in recent weeks. These must be addressed as a matter of urgency.
"For now the club's entire focus must be on using the remaining days of the transfer window to strengthen the team significantly."
Tim Payton, a spokesman for AST, was critical of Wenger's prudence.
"We don't know what's going on but clearly there's a manager that by his own words isn't a big fan of spending money on strengthening the team through expensive player acquisition," Payton said.
"His apparent obsession with being a developmental manager is in danger of holding the club back.
"I'm absolutely certain that Arsenal will invest in players before the window closes, but will they be players of the required quality we want to compete to win trophies rather than once again competing to come fourth?"