Lee, who damaged ankle ligaments just before the Chappell-Hadlee Trophy series, was initially to have a second round of scans Friday, but such is CA's desire to take him to the World Cup that the paceman will be tested on Tuesday, The Sydney Morning Herald said.
It is unlikely that pace spearhead Lee will be fit to play in the Caribbean, but after Australia's poor showings against the Kiwis, selectors are keen to give him every chance to make the trip, the newspaper said.
Given the severity of Lee's injury, and the fact that Australia have already selected injured Andrew Symonds in their 15-man World Cup squad, it was initially felt Lee would be ruled out of the tournament.
Stuart Clark is understood to be the likely replacement, should Lee be unable to compete in the World Cup.
Lee has been working with Cricket NSW physiotherapist Pat Farhart, and is now said to be walking without the assistance of a surgical boot, the Herald said.
Farhart declined to comment to the newspaper on Lee's progress, other than to say: "He seems to be going OK, but we're just taking it a session at a time."
The world champions Australia have lost their last five one-day matches -- two to England and three to New Zealand -- just a month out from the World Cup.
Australia has received good news regarding injuries to other key players with opening batsman Matthew Hayden confident after receiving scan results of his fractured right big toe.
Australian team physio Alex Kountouris believes Hayden will be available for the entire World Cup, despite initial fears that the fractured toe, sustained during Tuesday's last Chappell-Hadlee game, would limit his campaign in the Caribbean.
"The specialist informs us that the fracture appears to be stable," Kountouris said. "We're pretty hopeful."
Symonds has told Queensland team-mates that he is "feeling great" after surgery to his torn right biceps, and is confident of returning midway through the pool stage of the World Cup, the Herald said.
Michael Clarke (hip) and Ricky Ponting (back) have been managing their injuries over the past week, and are not thought to be in any doubt for the World Cup.