Woodward has spent Tuesday with the Scotland squad, who are preparing for Saturday's Murrayfield clash with Ireland in the Six Nations.
Scotland coach Matt Williams said: "We're delighted to welcome Sir Clive to our training session.
"It's a great opportunity for our guys to put their hand up and add to the claims they make in the international arena for a place in New Zealand with the Lions during the summer.
"They have been working exceptionally hard for the Six Nations and aiming to win for Scotland so to know that there could be the added honour of a Lions tour at the end of it is yet another incentive.
"However the main focus at the moment is facing Ireland this weekend."
Woodward cannot fail to have noticed the Scots' improved performance in France at the weekend, where they were narrowly beaten in a game they were expected to lose by a wide margin.
"I'm very grateful to Matt and the team for allowing me to come along and watch them train here at Murrayfield," Woodward said.
"I've also spent some time with Wales and I'll be joining Ireland too.
"It's great from a Lions point of view to be able to see all the players in a more informal setting - just observing training.