May picked up a hamstring injury during England's practice session on Tuesday in Vilamoura and he is expected to miss the first Test on Saturday week, further limiting the options on the wing.
Losing the 27-year-old will be a huge blow for the Red Rose with the back scoring 10 tries in nine matches this season after swapping former club Bath for Leicester Tigers.
Speaking about May, England forwards coach Steve Borthwick told Sky Sports News on Wednesday: "He picked up a couple of knocks in training this week, he's hurt his hamstring. It means he's likely to be doubtful for Argentina.
"Our medical team are very positive he will be fit for Australia. One thing I would say about Johnny - since I've got to know him working with the team, his preparation is meticulous.
"I've already seen it, just with everything he's doing to get back fully fit as quick as possible."
England are also set to be without Jack Nowell for all three internationals after it emerged the Exeter Chiefs winger will be on the sidelines longer than expected following a second operation on his fractured eye socket.
The 24-year-old underwent surgery shortly after suffering the fracture in the Aviva Premiership victory over Newcastle Falcons in early October but a bleed was discovered under the metal plate inserted under his eye, meaning a second operation was required.
Speaking about Nowell's chances, Exeter's director of rugby Rob Baxter said: "I can't see him having any involvement with England, especially now after that second operation. It puts things back a little bit and it is still a relatively sensitive area that he must ensure heals properly.
"You don't take risks around your eye simply because of the risk to your sight. He needs to be careful and I simply think the autumn internationals will be too soon for him."