Those complaints range from drifting at high speeds to steering wheel vibration and exaggerated motions from small steering inputs.
The NHTSA says that 363,000 Corollas from model year 2009 and an additional 136,000 from 2010 could be affected.
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Earlier this week the automaker announced that it was suspending production at two US plants. Two days ago the NHTSA requested documents from Toyota in an effort to determine if recalls were conducted in a timely manner.
The president of Toyota in Japan, Akio Toyoda, again faced media scrutiny for the third time in two weeks, pledging tighter safety controls and a return to the automaker's once-coveted safety record. However, he said he would not make the trip to the US to appear before Congressional hearings, unless required to do so.
At the press conference in Japan, according to the Associated Press, Toyota is weighing the options of a recall for Corolla, but has not yet made that determination.