The Lions head coach is considering Stephen Jones and Charlie Hodgson for the fly-half role against the triumphant All Blacks, whose victory on Saturday opened up an unassailable 2-0 series lead.
"In my mind, I'm not going to play him [Wilkinson] in the next Test match," Woodward told BBC Sport.
"We've lost the series and we've now got to look at playing either Stephen Jones or Charlie Hodgson next weekend."
Wilkinson had previously been given a "less than 50-50 chance" of being fit.
The England fly-half has suffered another so-called 'stinger' injury, which amounts to numbness in his shoulder and arm.
It is the latest blow to affect Wilkinson, whose post-World Cup career has been seriously disrupted by repeated fitness setbacks.
"He took a couple of stingers in the game that are going to take a while to settle down,' said Lions coach Ian McGeechan, after the squad arrived in Auckland from Wellington on Sunday.
"The medical staff are still assessing him, but I have to say there is a less than 50-50 chance he will be fit for next week. They are monitoring him closely at the moment."
Wilkinson's recent injury history suggests he could be sidelined for a considerable length of time, and questions will once again be raised about his long-term rugby prospects.
He was overshadowed by All Blacks fly-half Daniel Carter's masterclass during the Lions' 48-18 second Test defeat.
If, as seems inevitable, Wilkinson is ruled out of the tour finale, then Welsh international Jones would be favourite to take over, although Woodward knows he could do worse than look at an in-form Hodgson.
The Lions have already lost potential Test stars Brian O'Driscoll and Richard Hill to tour-ending injuries, and their problems might not end there, with centre Gavin Henson suffering from concussion.
"Gavin got a bang on the head in a tackle on (All Blacks wing) Rico Gear,' added McGeechan.
"Both players (Wilkinson and Henson) will continue to be monitored over the next 48 hours."
While Woodward ponders his Test selection for Eden Park next Saturday, McGeechan's midweek team face a tough encounter against Auckland on Tuesday.
The line-up once again includes players who can consider themselves unlucky to have so far missed out on Woodward's Test plans, with Hodgson, skipper Gordon Bulloch and full-back Geordan Murphy all entering that category.
Scottish international forward Jason White, who flew to New Zealand as a tour replacement last week, makes his Lions debut in the back-row alongside Welsh Grand Slam winners Martyn Williams and Michael Owen.
Scrum-half Matt Dawson, meanwhile, is handed a first start since the 19-13 defeat against New Zealand Maori in Hamilton more than three weeks ago, but prop Andrew Sheridan has withdrawn from the starting XV after aggravating an ankle problem, so Graham Rowntree deputises.
The midweek combination is unbeaten on tour, defeating Taranaki, Wellington, Southland and Manawatu, but they will have their hands full against an Auckland outfit seeking a hat-trick of Lions scalps following wins in 1983 and 1993.
Auckland, coached by ex-Samoan skipper and Northampton favourite Pat Lam, are likely to field several Super 12 players, but McGeechan wants another strong statement from his resilient group.
"What we need is a good week, on and off the field, and end the tour on a high winning note,' he added.
"You will not see any player going 'off-tour' this week - they are not those kinds of men."