Woodward has been gaining as much information as possible from former and current players, and Brennan's name has cropped up too often for the former England coach to ignore.
"If you speak to the players there's a huge amount of respect for the guy," Woodward told the Sunday Independent.
"Players tend to know better than most coaches. It's not just the Irish, but Welsh and English players as well."
The former Dublin milkman moved from Leinster to Toulouse in 2003 and immediately picked up a Heineken Cup winner's medal in an all-French final against Perpignan at Lansdowne Road.
"I haven't heard a bad word said against Trevor Brennan," said Woodward.
Woodward - renowned for his meticulous approach to coaching - is ensuring his preparations for the trip to New Zealand in June are as thorough as possible.
"I've spoken to quite a few players, and they probably don't know what they're actually saying when we're having these conversations," he said.
"But you talk about certain players and they'll say if they think they're up to scratch or that they don't want them in their team.
"I haven't heard a bad word said against Trevor, which, considering he has a pretty tough guy reputation, is, to me, impressive."