The 29-year-old, who has already notched 10 goals this season, is one of their most important assets, and the Clarets will want him fit and firing for the start of next season (August 14).
New Zealand's final pool match is on July 28, with the quarter-finals scheduled three days later. With build-up matches and a training camp, players will need to commit for up to a month and there will be concerns about injuries and Japan's sweltering summer.
The club will also be aware that Wood, like the rest of the Burnley squad, had barely three weeks break in the off-season, before this current compressed, hectic campaign, which finishes on May 24.
But Burnley will want to keep Wood happy, given his status and proven record, so his stated desire to be in Japan is important.
Oly Whites coach Danny Hay has been in regular contact with Burnley manager Sean Dyche.
"He is very supportive – which is a massive plus," Hay told the Herald on Sunday. "As an ex-player he recognises that Chris is getting a little bit older now and the Olympics is something you would like to look back on, as one of those big-ticket items.
"He's not the only person inside Burnley that might be involved in the decision, but obviously he will have a massive say. The number one thing is that Woodsy himself wants to be there, he is a proud Kiwi and wants to play for his country."
Hay hopes it could be a win-win situation, as Wood comes into the Olympics fold after a break, has three (or more) high quality games in Japan, before heading back match-fit and sharp.
Reid hasn't played for West Ham since March 2018 and battled a knee injury for more than 18 months. But he managed 12 MLS games with Sporting Kansas City, before joining Championship side Brentford in February in their push for promotion.
The 32-year-old is vital, with his ability to marshal the defence, along the belief he brings with his presence.
"Winston is desperate to be a part of it," said Hay. "We have sent West Ham a letter of intention about having Winston as part of the group and had no negative correspondence back thus far. Hopefully they will look favourably upon that."
The third over-age player is likely to be needed in central defence or right fullback, which makes Michael Boxall a prime candidate, though Bill Tuiloma or Tim Payne could come into consideration.
Hay will select 22 players, including four travelling reserves, and after "casting the net far and wide", has narrowed his focus to around 30 contenders.
Hay's other pressing concern is confirming warm-up matches, ideally three, ahead of the competition.
That unusually happens with the other competing teams, though logistics are still being worked through by the IOC, Tokyo organisers and Fifa given the Covid-19 challenges of squads arriving early.
Hay said NZF are looking at a training camp and games in Abu Dhabi or Qatar as a "plan B".
"That's an option, but I'm confident we will be able to play [in Japan] as the other teams will need games too," said Hay.
