Manchester United will push to secure David de Gea and Marcus Rashford to new deals before June.
The Daily Mail says negotiations are believed to be ongoing with De Gea, and preliminary talks have begun with Rashford over new deals.
The key pair are both out of contract in summer 2020, while midfielders Juan Mata and Ander Herrera - both free agents this summer - are also on the to-do list.
"We are a long way through the ones we had planned to do in this season," vice-chairman Woodward told investors.
"But obviously we want to finish off the final few as rapidly as we can."
Anthony Martial, Luke Shaw, Chris Smalling, Scott McTominay, Phil Jones and Ashley Young have all extended their contracts this season.